In order for the games to count, they can be done in either matchmaking or private matches. Ranked and casual modes both count toward the achievement, so choosing either works here. The four game modes are Rescue, Team Deathmatch, Cry for Help and Free for All. You can join a round in progress and get the achievement as long as you finish play to the match result screen.
This achievement has reported as glitching for some users and not unlocking. If it glitches for you and you manage to unlock it, please put a comment below so that I can update this solution to help out others.
5/6/13 Edit: Thanks for the tips from darthluigi7, Ravage NZ and WEBHEAD 7 this is confirmed to be working in private matches.