In Awesomenauts, assassins are 'nauts who specialize in ambush, killing isolated or vulnerable targets, or putting them in such position in the first place. They usually have some form of escape tool and good initiation as well.
The "job" of an assassin usually consists of approaching the enemy team using tools that allow them to do so from outside their view and/or by giving them little time to react, (examples include invisibility, hide areas, flight, etc.) to pick a target and isolate and/or burst it as soon as possible, so they can make an escape.
In a team environment, they can also use an initiation to help their teammates in a teamfight. Assassins tend to put themselves at risk in order to do their job, which is why they have escape tools.
Leon and Vinnie & Spike are considered assassins by the game. Leon's abilities have upgrades that may change the way he assassinates targets: for example, a Leon who purchases Surprise Party Mask will typically use cloak to initiate on his target, while a Leon who doesn't may use it to escape.
Similarly, Vinnie & Spike may Dash against an enemy target or even a team, and then use the Smoke Screen upgraded with Red Bandana to escape. These are just examples of how much variety can be achieved in the game.
Although the aforementioned 'nauts are considered the assassins in game, the role is not set in stone: any character may use assassination techniques to kill an overextended or vulnerable target. Perhaps the most notable example of this is Clunk using his "divebombing" technique to surprise and kill his enemies. It doesn't make him an assassin, but it's an example of a 'naut playing a different role to some extent.