Crysis 2

There is a total of 6 game modes you have to win atleast one match in each.

Team Instant action - Unlocked at rank 1. This gamemode is basicly a 6v6 team deathmatch the first team to reach a total combined kills of 50 or the most kills at the end of the time limit is the winner.

Instant Action - Unlocked at rank 1. A free for all gametype with 12 players first to 25 kills wins.

Crash Site - Unlocked at rank 6. Objective game mode 6v6, Secure the alien drop pod and hold. Earn points by holding (standing next to the drop pod) the longest first to 150 points wins.

Capture the relay - Unlocked at rank 12. Objective game mode 6v6. Basicly a capture the flag game mode , grab enemy relay and bring it back to your base to score. Team with the most captures wins the match.

Assault - Unlocked at rank 18. Objective game mode with only one life per a player 6v6. Two sides one attacking and one defending. Team attacking must infiltrate enemy teams bases and download data from points a,b,c,d, and e. (less points depending on amount of people on teams) Defending team must stop attacking team from dowloading 100% of the data or kill entire enemy team. Attacking team must kill all defending team or dowload 100% of the data.

Extraction - Unlocked at rank 22. Attacking team must take and capture Two alien bio ticks. Defending team must protect the two ticks from the attacking team. Attacking team wins by captureing both ticks and defending team wins by stopping the attacking team.

This achievement is unlocked by winning one game in the 6 different game types the game offers (team instant action, instant action, crashsite, capture the relay, assault and extraction).
The last gametype is unlocked on lvl 22 but you can also join a friends lobby if you are on a lower lvl and want to get the achievement.

Note: you just have to win the normal game variant and not the special ones (eg.: hardcore and classic) for the achievement to unlock

if you want to know which gametypes you have already won a game in it can easily be checked by watching the agency file

A glitch discovered by the TA user Sudden Fall allows you to start a public match by yourself which means that you don't need a party of 6 anymore.

You do need to be level 22 however to be able to access the last game mode available.

Sudden Fall made a great YouTube video describing how the glitch works. You can check it out here.

For those having a hard time to pull of the glitch: the key is to keep switching between public and private modes fairly quickly until the game gives you the option to start a public match by yourself. I recorded this clip to show you how it's done. I hope it will help.

Now that you know how the glitch works just start a public match in the game mode of your choice by using the glitch and let the timer runs out once in the game. Rinse and repeat for the remaining 5 game modes and the achievement will unlock.

You will also unlock the League of Your Own achievement if you haven't already after completing your first match as you were the only player in the game.