Take a look at this video which explains it all with a few demonstrations (1a is also known as single a, 2a is also known as double a and 3a is also known as triple a.)
Here are a few demonstrations of yoyo contests with those styles of play:
The 1A/2A/etc naming convention dates back to when yoyo contests only had 1A and 2A divisions. When the modern world contest started, the top players all competed in the 2A division. The 1A division was considered a lesser division and was more for fun, and 1A did not officially crown a world champion like 2A did (kind of like ladder/pick-your-trick competitions in today’s contests). The progressive naming probably represented the increased difficulty going from 1A to 2A (similar to how high school sports sometimes divide up into class 1A, 2A, etc based on school size), and may also have reflected that the divisions used 1 and 2 yoyos, respectively. At the very least, it made sense as a mnemonic that 1A was 1 yoyo, and 2A was 2 yoyos.
When the 1A division became more advanced, it started to split into different styles (most notably offstring and freehand), which still competed under 1A since they were 1 yoyo styles. In 2000, the world contest added an X division to represent any style outside of the two competition divisions, and the 1A offshoots moved there (along with 3A). After a few years of X division, that was split into three distinct competition divisions, and they just decided to extend the 1A/2A naming convention to cover the three new ones, even though it ended up being arbitrary.