FFA Mode (which stands for Free-For-All) is one of the 5 currently available modes in the game and was the first mode avilable when Agar.io was released. In this mode, you are on your own against all the other players in the lobby. Teaming is possible, though, and teams usually dominate the FFA servers, even though they might trigger the anti-team penalty.
Your objective is to grow and become number 1 on the leaderboard (and try to stay there). The places on the leaderboard are determined by the total mass of the player's cells. You can consume any other player in the lobby.
In this mode, you can use Skins, although you will still have a colored outline around it.
Occasionally, a server will stop respawning cells and instead spawn new cells in another server. This leads to a last-man-standing style of deathmatch among the remaining cells, as any cell that is eaten will leave the server (pretty much elimination here). This deathmatch will last roughly 10-20 minutes, at the end of which all cells will be transferred to another server, alive or not. Generally, only cells in the top 10 will remain alive to compete in the deathmatch. You can find more info about this on this page.