To begin with, there are (currently) only 5 game modes, each of which is pretty standard if you've played similar FPS games in the past.
The game modes are: Escort
move a target between checkpoints while the other team defends and then switches sides
Domination
three different points on the map must be conquered to earn points for victory
Occupy
capture a single point to gain points for victory, the point moves every minute
Zone Control
The attacking team must take control of 5 different points while the defending team tries to prevent this, after one round the teams switch sides
Hot Shot
Each kill drops a bounty, if you have the most bounties you become the hot shot, you move faster and collect bounties if the hot shot has more points to win. However, your location will always be shown to the opposing team.
To get this achievement, you must play a match in each of the above modes. Winning or losing does not matter. I can also confirm that entering a match during the game counts.
Note:
If you signed up for the Welcome Playlist, which the game recommends, you will not have access to Hot Shot or Zone Control, so make sure you exit the queue and queue specifically for those two modes at some point after playing one of the other three modes from the Welcome Playlist.