It's similar to a sphere, but it's an Emanation so it extends 10 ft away from the character instead from a point in space.
Level 6: Aura of Protection
You radiate a protective, unseeable aura in a 10-foot Emanation that originates from you. The aura is inactive while you have the Incapacitated condition.
EDIT:
Emanation [Area of Effect]
An Emanation is an area of effect that extends in straight lines from a creature or an object in all directions. The effect that creates an Emanation specifies the distance it extends.
An Emanation is an area of effect that extends in straight lines from a creature or an object in all directions. The effect that creates an Emanation specifies the distance it extends.
An Emanation moves with the creature or object that is its origin unless it is an instantaneous or a stationary effect.
An Emanation’s origin (creature or object) isn’t included in the area of effect unless its creator decides otherwise."
The effect extends for a specified distance from the creature - in the case of a paladin aura below level 18 this distance is 10' - however, the definition of an emanation does NOT use the word radius.
So one interpretation is certainly a 10' radius circle.
However, the rules for areas of effect have some odd interactions with the rules for playing on a grid:
"Ranges. To determine the range on a grid between two things—whether creatures or objects—count squares from a square adjacent to one of them and stop counting in the space of the other one. Count by the shortest route."
In this case, a 10' emanation ends up being a 5x5 square since two squares in any direction from the center are all considered to be within 10'.
This is the usual approach used for 10' reach weapons as well as other reach abilities - just count the number of squares. I also know DMs who use the same approach for measuring all distances when playing on a grid including areas of effect (because it is RAW and because they also find it simpler to adjudicate). Other DMs require about 1/2 of the square to be within the area of effect to be affected (which would leave out the corner squares of a 5x5 for the paladin aura. DMs who run on hex grids will have other adaptations.
So, the actual answer to this question is ask your DM since if you are playing on a grid, RAW, each square counts as 5' and the paladin aura will be a 5x5 group of squares. Many DMs choose to go with the 10' from the creature approach and use a 10' radius circle even when playing on a grid. This is arguably not RAW for play on grids but many if not most DMs might go with the 10' radius approach.
In addition, I would note that the definition of an emanation has removed the word "radius" that was present in the 2014 version of the rules and instead just specifies the 10' distance making the rule more compatible with the rules for playing on a grid should a DM choose to use them.
P.S. If you look further at areas if effect, a "sphere" extends an equal distance in all directions from a point in space while a "cube" extends an equal distance from a point on the face of a cube. The difference is the relationship between the point of origin and the resulting shape and not so much whether the actual shape is a "cube" or a "sphere" ... e.g. a sphere could be interpreted as a volume centered cube while a cube could be a face/surfaced centered cube depending on whether the rules of playing on a grid are in effect.
If any part of a square is under the template, that square is included in the area of effect. If a creature's miniature is in an affected square, that creature is in the area. Being adjacent to the edge of the template isn't enough for a square to be included in the area of effect; the square must be entirely or partly covered by the template.
So in my case, an Emanation should include and affect the following squares. For example, for Spirit Guardians:
But that's my personal way of doing it.
That doesn't mean it's hard for me not to see an Emanation as similar to a Sphere when both AoEs have similar wording.
Sphere: "A Sphere [...] extends in straight lines from a point of origin outward in all directions. [...]"
Emanation: "An Emanation [...] extends in straight lines from a creature or an object in all directions. [...]"
A Cube is different because it states: "The effect that creates a Cube specifies its size, which is the length of each side."
As a side note, an interesting case can be found in a spell like Ice Knife, where the text states: "The target and each creature within 5 feet of it must succeed on a Dexterity saving throw or take 2d6 Cold damage". In this case, IMO, it should cover this area, even without using the XGtE rules:
Is the paladin Aura of Protection a Sphere or cube or something else ?
It's similar to a sphere, but it's an Emanation so it extends 10 ft away from the character instead from a point in space.
EDIT:
Something like this when playing on a grid:
The actual answer to this is to ask your DM.
The rules for an emanation state:
"Emanation [Area of Effect]
An Emanation is an area of effect that extends in straight lines from a creature or an object in all directions. The effect that creates an Emanation specifies the distance it extends.
An Emanation moves with the creature or object that is its origin unless it is an instantaneous or a stationary effect.
An Emanation’s origin (creature or object) isn’t included in the area of effect unless its creator decides otherwise."
The effect extends for a specified distance from the creature - in the case of a paladin aura below level 18 this distance is 10' - however, the definition of an emanation does NOT use the word radius.
So one interpretation is certainly a 10' radius circle.
However, the rules for areas of effect have some odd interactions with the rules for playing on a grid:
"Ranges. To determine the range on a grid between two things—whether creatures or objects—count squares from a square adjacent to one of them and stop counting in the space of the other one. Count by the shortest route."
In this case, a 10' emanation ends up being a 5x5 square since two squares in any direction from the center are all considered to be within 10'.
This is the usual approach used for 10' reach weapons as well as other reach abilities - just count the number of squares. I also know DMs who use the same approach for measuring all distances when playing on a grid including areas of effect (because it is RAW and because they also find it simpler to adjudicate). Other DMs require about 1/2 of the square to be within the area of effect to be affected (which would leave out the corner squares of a 5x5 for the paladin aura. DMs who run on hex grids will have other adaptations.
So, the actual answer to this question is ask your DM since if you are playing on a grid, RAW, each square counts as 5' and the paladin aura will be a 5x5 group of squares. Many DMs choose to go with the 10' from the creature approach and use a 10' radius circle even when playing on a grid. This is arguably not RAW for play on grids but many if not most DMs might go with the 10' radius approach.
In addition, I would note that the definition of an emanation has removed the word "radius" that was present in the 2014 version of the rules and instead just specifies the 10' distance making the rule more compatible with the rules for playing on a grid should a DM choose to use them.
P.S. If you look further at areas if effect, a "sphere" extends an equal distance in all directions from a point in space while a "cube" extends an equal distance from a point on the face of a cube. The difference is the relationship between the point of origin and the resulting shape and not so much whether the actual shape is a "cube" or a "sphere" ... e.g. a sphere could be interpreted as a volume centered cube while a cube could be a face/surfaced centered cube depending on whether the rules of playing on a grid are in effect.
I don't disagree completely, as I use the rules from XGtE (p. 86, see below) and I don't use the optional rule in the DMG about diagonals on a grid, where range is measured differently.
So in my case, an Emanation should include and affect the following squares. For example, for Spirit Guardians:
But that's my personal way of doing it.
That doesn't mean it's hard for me not to see an Emanation as similar to a Sphere when both AoEs have similar wording.
A Cube is different because it states: "The effect that creates a Cube specifies its size, which is the length of each side."
As a side note, an interesting case can be found in a spell like Ice Knife, where the text states: "The target and each creature within 5 feet of it must succeed on a Dexterity saving throw or take 2d6 Cold damage". In this case, IMO, it should cover this area, even without using the XGtE rules: