How do you choose him and what are his roles?

You can have one, but we don't bother. Usually the party is in harmony as to what they do, so we're more egalitarian. I've only ever had a persistent argument where one side didn't fold after a brief discussion, which we resolved using a contested Charisma (Persuasion) check. 

I'd be uncomfortable with a leader because that puts one player above the others.

It really depends on the group.  For some groups, particularly with newer players or a group of chaotic-X characters, having someone to guide them in the right direction can be really helpful. This need not be the leader per se, but could be some kind of lawful (good/evil/neutral irrelevant) or otherwise moralistic good character who pushes them in a unified direction - more the party’s moral compass than an actual leader.

In other groups everyone can work together without a leader at both the player and character level and one singular leader is not necessary.  In these types of group, someone trying to establish themselves as a leader tends to be the result of that player having Main Character Syndrome - wanting to make them the centre of the story and not respecting their other players - which can make the game worse for everyone else. 

In groups where a leader is helpful, they do not need to be the face, though it helps for them to have some charisma to excel at convincing their group to do something if there are opposed checks needed. 

We don’t have a formal leader, but we do have someone who kind of leads decision making. Basically they are the one willing to say their opinion and get conversations moving. But they don’t make the decision, just get the ball rolling. But that’s a player who does that, not a character.

And if there were to be a leader in character, I’d hope it would be someone with a high int or wis. If you base it on cha, you could get lead into some really bad choices. Seems more like a demagogue or cult territory. 

I think, in most groups, a leader isn't chosen, but agreed upon by the group as they progress in their adventures.  One player (or character) will, as stated above, more or less "take command;" in many situations, leading discussions or sorting ideas and rehashing them as they develop.  That person (or character) may or may not be the face of the party, perhaps the Fighter makes most of the decisions and presents most of the ideas, but the part gets the Sorcerer to do the face work, persuading guards or officials, negotiating contracts and such.

I think most groups will end up being highly diplomatic and discuss and agree on major plans.  For less important things, one person (or character) may well "lead" but on a whole, the group will operate more as a group than a team with a leader heading it.  I know our main group has no leader, per say, but for in-group discussions on what we are going to do or what plan we want to employ, the group looks to my Monk often as the one showing wisdom, thoughtfulness and a variety of angles to each situation.  Our secondary troupe, my Barbarian has taken the role of leader, being brash and commanding, but has been twice "talked out of" ideas by the others.  I feel that my Monk, whom I play with similar aspirations and ideals to my own, nets more respect and trust in his decisions that the other guy, who isn't much like me at all, lol.  It might also help that I am, IRL the dad of 2/4 of the others in my group.

In most games I have played having a leader or one who makes basic decisions for the group is very important and speeds game play. The leader position can change depending on the situation, such as the face may lead during social encounters, the thief may lead during encounters with lots of traps, etc. 

I have played in games where most if not every decision was a discussion and it greatly slowed the game and often ended in the strongest personality deciding what, where and how the group was acting.  

In my group, most of the characters have very specific “jobs” for example, my character manages treasure and group money. But for most major decisions, we usually vote on them.

 In most cases, it isn’t very beneficial to have a group leader, as they will start to feel like the main character, and other players might get jealous.

Quote from Skfma >>

How do you choose him and what are his roles?

 Really depends on the setting.

In one of my games I am running (with six players) - the human paladin is the "face" of the party. He's usually the one who might ask townspeople for directions and such. But deciding things has always been a group (out of character) type discussion, before roleplaying choices.

And that could change; for example where they are in my game (homebrew world), it's mostly humans who do not care for "others" (Elves, Dwarves are fine; but things like Tieflings, Dragonborn, are looked at suspiciously - and as such have Disadvantage because of their stereotyped fear for your common citizen). But should there go somewhere else, where perhaps Dwarves are the main setting, someone else might step up to be the "leader."

In my other game (only three players), there is no leader at all. They all kind of talk things through (much easier with so few players).

And in the other game, there's two monks, who tend to be the leader of the group (six players there too), even though technically the tiefling bard has the highest charisma by far.

Quote from ShhinySilver >>

 In most cases, it isn’t very beneficial to have a group leader, as they will start to feel like the main character, and other players might get jealous.

 

 This is why often I recommend rotating the position. Also this does not prevent voting on significant issues in the game if the situation allows for it, ie in game vs out of game time.

In general it can be harder to rotate other positions such as trap finder/remover, face, etc but in general it is easier in 5e to have multiple classes fit such roles then in past editions.  

I'm a forever DM, and my group of old players never have a formal leader (unless the current campaign calls for it). But one of them usually assume an informal leader role, and as he also is very good at subtly dividing the spotlight and make sure every character get their focus time, all players are happy with it. As it also takes a bit of load off my shoulders, so am I :-)

It is important to NOT have a leader--unless you're looking for a specific kind of role-playing that involves internecine conflict. Notably, the choice of gender pronouns in the question points to one of many incipient disagreements. It is a strength of 5e that it strives for emergent storytelling. Choosing a character to be "leader" will limit as much, or more, than it will help in that goal. Instead, look for leadership from players as their characters engage with scenes. The leader will be the character and/or player who is most invested.

We like our party leader because she is great with strategizing and sets the right amount of chaos into the story when she negotiates on the party's behalf. She also never tries to steal the spotlight from anyone else's story unlike some people we played with in the past. It might also help that her story is an amnesiac—only remembers things that happened since one month before meeting the party. Anything can happen, and it could easily be something she did but doesn't remember. "It's not my fault... I think." is something her character says a lot.

I often find a “leader” kinda just emerges on their own, but a good leader makes sure to lead the party to make decisions together instead of just taking charge. This is usually a player role though, not a character role, so it happens independently of a particular party role.

Quote from Caerwyn_Glyndwr >>

It really depends on the group.  For some groups, particularly with newer players or a group of chaotic-X characters, having someone to guide them in the right direction can be really helpful. This need not be the leader per se, but could be some kind of lawful (good/evil/neutral irrelevant) or otherwise moralistic good character who pushes them in a unified direction - more the party’s moral compass than an actual leader.

In other groups everyone can work together without a leader at both the player and character level and one singular leader is not necessary.  In these types of group, someone trying to establish themselves as a leader tends to be the result of that player having Main Character Syndrome - wanting to make them the centre of the story and not respecting their other players - which can make the game worse for everyone else. 

In groups where a leader is helpful, they do not need to be the face, though it helps for them to have some charisma to excel at convincing their group to do something if there are opposed checks needed. 

Agreed, it really depends on how the group synergizes together, they may naturally come to a leader (as Sposta suggested), they may pick one at the start, or, they can find that they just don't need one.

No campaign or adventuring crew needs a party leader, if it works best for your group, you can have one. But if each character is able to plan and tune on their next courses, and agree on a course of action without endless infighting, then a group "leader" may never be necessary.

I've found that it's unlikely for players of D&D 5e to have an official party leader in their group, but usually there ends up being one player who is better at settling decisions than the others. For example, in my current campaign we often come across situations with a lot of different options on what to do next, with all the characters tossing out ideas on what to do next and concerns about potential repercussions, and this can sometimes lead into a sort of feedback loop where nobody can settle on what to do and sometimes we lose track of what to do. I usually end up being the one to step forward and narrow our options down to just two things or occasionally just pick the one option I feel is best, since despite playing the Barbarian of the group my character is probably the most level-headed. He's not necessarily the "leader", but he has been referred to as the "Team Dad" several times.

However, the character isn't really the Face of the party. We have both a Bard with insanely high Persuasion and a Sorcerer with great deception, and those two usually take front stage in social situations. If we need to interact with the public my character, at best, tries to offer some advice or a few words of encouragement to provide the help action for my allies, but I'm only the one making the charisma checks if they're not able to for one reason or another.

Depends on player make up, but like Sposta and Transmorpher mention, sometimes a leader or decision making facilitator arises.  You could also have situations where "leadership" "shifts" depending on what the party is doing.  

Some groups really need a leader, I find. Too many ideas, too little focus, too much indecision...having a character that inspires action can really keep things moving. As long as the player (and character) isn't monopolizing the game, it's generally fine. That said, I've never actually seen a "face" character in a game who is the leader. I'm sure it happens, but in my experience, the smart, wise or gutsy characters frequently find themselves in that position. 

My mouthy fighter ended up the party leader over the centuries-old paladin, likeable warlock and friendly bard because she was the strategist and moral center of the party. 

In another party, our barbarian is the leader because he's got a heart of gold and the most courage of us all.

In the campaign I run, the trickster cleric has become leader because he's cunning and comfortable with risk. 

When party leaders emerge, they're often the ones who are respected within the group and have clear-cut ideas about what needs to be done - or at least what is right. These are characters who pick a direction and believe in it strongly enough that everyone else follows. That direction can range from "Hey, this is fun!" to "We must right this wrong," but they sell that belief to the others. Note that this is different from stubborn characters who just bully or Leroy Jenkins the party into following them. Those aren't leaders. Those are hostage-takers, and they tend to sour the game for everyone until they're reined in. 

As someone that has run 8- 10 players for the better part of a decade, you NEED leaders at the table. Whether you have a separate one for combat, social, and exploration pillars or you have a single real life player that "gets it", pays attention, takes copious notes, and isn't afraid to decide and speak up - it's a must have at our table. I've spent half a century among high-level and highly-skilled leaders. I've witnessed leadership at the highest levels, where life and death is at stake. I've seen companies fall, some come and go, and been part of groups, seminars, events, etc., with a good leader, with a bad leader, and without any leader at all. At the end of the day, D&D is a group of people trying to have an amicable, exciting, and worthwhile investment of time and resources, all handled through imagination and moving some toys around. An endeavor of that sort always has and will forever function better with a leader.

Need more proof as to the importance of leadership. It's why a role called dungeon master exists. It's never been 7 players. It's 6 players and one DM.

Leaders are emergent in all human social groups. 

You will always get them, whether you want them or not.

And in my experience, when you let them emerge organically, you get much better ones than when you deliberately create them through some artificial process.

I think having a leader can cut through analysis paralysis when a party is discussing options. It's good to have someone who can listen to the discussion, and when they feel it looping back in on itself or getting lost in minutae, they can interject with "ok we've got a lot of good ideas, we should pick A or B and do that." I find it just keeps the plot moving. 

I don't think this person has to be the party face, as you can always direct the high charisma character where to go and who to butter up/intimidate.