Cobalt offers many different Gamemodes to choose from while playing. They make the most fun when played with a friend and a second controller, but can also be played alone. You can see a list of all official maps classified by gamemode here.
As of v123, 3 kinds of challenges exits, each considerated a 'different' gamemode. To install user-made challenge maps place them on documents\cobalt\maps (with the rest of user-made maps you have installed). Note that since they don't use a custom prefix, and neither are they on a 'special' folder, no gamemode icon/name will appear on the user maps menu list, but they will still be playable.
Combat challenges can have different objectives, but all rely on a 'target' system. Usually, you'll need to free prisioners or kill certain enemies to complete the map, but if you run out of time from the top-right corner timer, you will fail the challenge.
So far puzzle challenges are an unknown type of challenges.
On speed challenges, the timer is all that counts! Get to the end of the map using the less time possible! (Note: You must reach the end alive!)
In Deathmatch, you win by scoring a certain number of kills before any of the enemies does so.
The old version of the deathmatch gamemode was played in teams only, and while it didn't have the new features that Deathmatch can offer, it allowed for a lot more customization in terms of spawning loadouts and loot drop.
You can still install community-made maps that use this gamemode.
In Team Strike, two teams composed by 4 members fight each other during various rounds. The one who wins most rounds is the winner. Several pre-build 'classes' can be bought, sold or repaired between every round. If the player dies, he can take control over a bot on his team.
In Capture the Plug, you have to gather the enemy team's plug and bring it back to your plug station while your plug is also at your base, so you can score a point.
As of v123 Alpha, Survival was completely revamped. Survival now counts with a dynamic shop, gambling, smaller but more intense maps, and enemies are fought in waves. Most maps also contain a boss that will spawn at a certain wave.
The original Survival mode. More adventure-like, with all shops being physical, hidden secrets, (usually) bigger maps, and an endless duration.
In Bounty, your goal is to be the first to reach a defined number of volts by collecting loot and bringing it to the converter. As of v123 Alpha, bounty isn't included in the game, since, according to the devs, they are looking into how to revamp bounty too.