There are 5 different styles of yo-yo play which I will detail below - but everyone starts off with 1A:
1A - Also known as Single A or 'One Handed Yo-yoing', this is the style that everyone starts out with and is by far the most popular in both the competitive and casual yo-yo world. This is simply one yo-yo attached to your finger, performing tricks from Walk the Dog to Split the Atom.
2A - Also known as Double A or 'Two Handing Yo-yoing' - where by a player has one yo-yo in each hand and performs Looping style yo-yo tricks. Proficient two handed looping has always been a sign of a yo-yo master. This style is renowned to this day as being the most difficult style of play to master.
3A - Also known as Triple A is where the player uses a yo-yo in each hand, but this time instead of performing looping style tricks (as per 2A) they will instead perform string based tricks (much like in 1A). This style is also considered one of the most difficult to master.
4A - Also known as Off-String is, as the name implies, where the player uses a yo-yo that is not attached to the string. A bind is used to wrap the loose string around the axle, then the yo-yo is throw and caught on the string (much like a diabolo). Generally players use much larger than normal yo-yos for this style of play, and they are usually made out of very durable materials such as rubber or certain types of plastic, this is to ensure they are robust enough to withstand repeated drops on the floor during practise.
5A - Also known as Freehand or 'Counterweight', this style of play involves the yo-yo string not being attached to the finger, and instead to some form of small weighted object (usually a rubber ball or casino die). This allows the player to let go of both the yo-yo and string entirely.