I have heard a lot about player development and downtime activities, but I don't really understand them. Can someone explain to me how these things work, and how I might implement them into my adventures.

Your players just finished crawling through a dungeon and returned the McGuffin to the quest giver. They now have a day/week/month's of downtime before they are going to be summoned again for their next quest. This is the behind the scenes things the characters are doing when the players aren't actively controlling them.

There's tables in the Dungeon Master's Guide and in Xanathar's Guide to Everything that you can have the players roll on during this downtime. You ask the player what they want their character to accomplish. This may be crafting items, running a business, building a stronghold, just gambling away their hard earned cash. Whatever they're thinking. You find the table, they roll the dice. 

They may want to learn a new skill which, by default, takes 250 gold and 250 days (RAW, adjust to fit what's best for your group). Or may want to do crafting, which I would recommend cutting the time in at least half of what's displayed in the Dungeon Master's Guide. 

They may also just want to not do anything at all so they would just deduct standard cost of living (found in the Player's Handbook) for the period. 

So, first off, reference the DMG (if you have it on here, https://www.dndbeyond.com/sources/dmg/between-adventures#DowntimeActivities), and the passing mention in the PHB (https://www.dndbeyond.com/sources/phb/adventuring#BetweenAdventures).

The new Unearthed Arcana is a playtest version of one kind of Downtime activity, called a Bastion, which is basically a "base of operations" for a character or group of them.

The game is designed to use the time between adventures as a way to suck some money out of the PC's hands -- There is a cost for their lifestyle, and they deal witht hat in Chapter 5 of the PHB.

Beyond that though, there are a lot of different things that might go on: researching new spells, crafting stuff, making an ordinary living, researching, training, and all the stuff that isn't "adventuring", including things like stocking up on supplies and such.

The default rules don't give much detail because for the most part, it is all handled by a couple lines about "what you did before this next adventure" in many games, maybe with a couple rolls, and is generally handled in a few minutes. 

For some folks, however (such as I) "downtime" is also when they get to advance in level, recieve rewards, transcribe spells they found (because I come from an older version of the game where you are supposed to find your spells if you are a mage), and do all of the stuff above. For my games, we actually role play those things out to a certain degree. But I want to stress that this is not the norm, and we have to step outside the normal rules to do it.

So let's say you completed Phandelver and Below, and the party heads to Neverwinter afterwards.  you know that the next adventure is going to be Dragon of Icespire peak. So, what happens between the end of Phandelver and the start of Dragon?

That's Downtime. The stuff that happens in between adventures. this period is about EVERYTHING that isn't going out to adventure. And you can role play and engage with it as much or as little as you want.

The game assumes you will do very little of that. Even the most hardy of traditional heroes stops for a time to rest, celebrate, and attend to the "boring daily stuff". It is the adventuring equivalent of "going to the bathroom" -- you know folks do it, but it isn't really covered by the rules, lol.

So, really, it is whatever you, as the DM, want it to be -- "an adventure without combat", or just the stories of how they talked to their teacher and went from 1st level to 2nd level.

As someone who still thinks in terms of a live table game, with folks sitting in chairs around it, downtime is also a way for the story line lovers in the group to have a little fun between table sessions, without slowing down table play for those who have less interest in such things. If done via phone or email [old school methods], or through something like Discord,  a few minutes to maybe an hour of one-on-one with the players can help them feel more connected to this world. We only do day to day type stuff that the PC can do without getting involved in combat, or somehow single-handedly causing a shift in the plot line. [Note: This can be done via messages, over time. One does not have to arrange for dedicated blocks of time, per player.]

When I am the GM, I offer these between table session interactions to all players in the party. Each player generally has at least a week in real time, between table sessions, to take part at their leisure, or not. Some take me up on it, and some don't. For those who choose not to take this opportunity, I provide a minimum amount of table time for them to get their downtime stuff sorted: "OK, what do you want your character to have accomplished over this week?", and make some rolls, depending on the reply. If that player starts wanting to go deeper into character, I say something along the lines of, "I offered to do this for you, in detail, during downtime. The rest of the party is ready to go, now."

If most or all of the players want to do all of this at the table, that's fine, too. They just need to be OK with spending bigger chunks of table time in RP, and less time in action/combat.

Just presenting play options.

I never really understood the true rules of crafting. Is there some kind of D&D guidebook of recipes with materials the players need to collect before crafting, or do the players need a special ability to craft each unique type of item. 

Quote from RJOP >>

I never really understood the true rules of crafting. Is there some kind of D&D guidebook of recipes with materials the players need to collect before crafting, or do the players need a special ability to craft each unique type of item. 

 
There is very, very little on crafting at all.

The bulk of it is in Xanathar's, and they do not give recipes or formulas.
https://www.dndbeyond.com/sources/xgte/downtime-revisited#CraftinganItem

What they do instead is say "go kill a creature within this CR range", and then give you a few examples:

If appropriate, pick a monster or a location that is a thematic fit for the item to be crafted. For example, creating mariner’s armor might require the essence of a water weird. Crafting a staff of charming might require the cooperation of a specific arcanaloth, who will help only if the characters complete a task for it. Making a staff of power might hinge on acquiring a piece of an ancient stone that was once touched by the god of magic — a stone now guarded by a suspicious androsphinx.

In addition to facing a specific creature, creating an item comes with a gold piece cost covering other materials, tools, and so on, based on the item’s rarity. Those values, as well as the time a character needs to work in order to complete the item, are shown on the Magic Item Crafting Time and Cost table. Halve the listed price and creation time for any consumable items.

So their recipe boils down to:

Figure out what you want to make.

Make sure you have the skill to do it (through your ability skills, a tool set, or a kit you bought).

Pretend you got a recipe for it. If it is magical, it always requires some part from some monster. Y

Your DM will tell you what the monster is depending on what you are creating.

You go kill the monster, collect the part, return and start making it.

It takes time to make something -- magical stuff always takes a long time, non-magical stuff is pretty fast.

You spend the gold for it, you make a die roll against a DC your Dm sets, and you are done.

This is normally handled as just a conversation. 

THe 5e crafting system is notoriously blah.

Quote from first_rolled_a_die_in_1976 >>

As someone who still thinks in terms of a live table game, with folks sitting in chairs around it, downtime is also a way for the story line lovers in the group to have a little fun between table sessions, without slowing down table play for those who have less interest in such things. If done via phone or email [old school methods], or through something like Discord,  a few minutes to maybe an hour of one-on-one with the players can help them feel more connected to this world. We only do day to day type stuff that the PC can do without getting involved in combat, or somehow single-handedly causing a shift in the plot line. [Note: This can be done via messages, over time. One does not have to arrange for dedicated blocks of time, per player.]

When I am the GM, I offer these between table session interactions to all players in the party. Each player generally has at least a week in real time, between table sessions, to take part at their leisure, or not. Some take me up on it, and some don't. For those who choose not to take this opportunity, I provide a minimum amount of table time for them to get their downtime stuff sorted: "OK, what do you want your character to have accomplished over this week?", and make some rolls, depending on the reply. If that player starts wanting to go deeper into character, I say something along the lines of, "I offered to do this for you, in detail, during downtime. The rest of the party is ready to go, now."

If most or all of the players want to do all of this at the table, that's fine, too. They just need to be OK with spending bigger chunks of table time in RP, and less time in action/combat.

Just presenting play options.

Yeah I agree that its more of an out of table thing. 

I try to use downtime as something to generate small encounters  as distractions if needed or to give things back story.  I don't worry about when the down time actually happens that much I just say you have x amount of down time  they tell me what they want and what they roll, I roll on some tables  and tell them what happened they respond or I run a  small combat or something  in the next couple of sessions. 

So making an item I'd be like 

1. I ask them to come up with a method of how to make a magic item. It needs to make sense for the item. So maybe they ask for a dragons help with a flame tongue or something 
2. I check for a complication then come up with some rolls and possibly an encounter. Usually they have to find what they need with some kind of knowledge check then convince some one or fight.
3. I run the mini encounters as distractions when needed. E.g I run a combat when there is lots of talking, run a talking when there's lots of combat ect.. other wise its just back story I ask the player to share if relevant.
4. if they do a really good job they maybe get some minor properties or some thing added on

 Training it would be similar. They come up with an idea who they want to train or how they want to train, they do some rolls, I maybe roll for a complication ect... 

Quote from RJOP >>

I never really understood the true rules of crafting. Is there some kind of D&D guidebook of recipes with materials the players need to collect before crafting, or do the players need a special ability to craft each unique type of item. 

 I just get players to make up with a recipe. They need special materials and some special way to craft it that is sufficiently magic and suits what they are making. If I want to use a specific recipe its usually part of a quest in which case it depends what I want them to do.  If I want them to  slay a dragon I can find something from a dragon or it's hoard to make pretty much any magic item. 

I dont find specific recipes that helpful because then I just have to shoehorn in what they need. Its easier to fit the recipe to what I want.

downtime is an away from the table accounting trick. At least for adventure league.  Hey DM I burned 15 downtime days and made 3 scrolls. or

Bob dropping a pound of maps and notes on the dm at the end of session. 'Hey DM. I got the design of my castle and town. Next week can we go over them." Three hours later the DM and Bob escape the trunk of his pinto. Next week.

DM, "9 months have pass in game and Bob has been boring you with the troubles he had of getting pink marble for his kitchen counters. i going to start the opening scene of the Temple of Doom  module. Your party is running from a boulder....."