A
MOBA is basically DOTA-like games. The goal is to destroy the enemies base. You play as a extra strong character and fight the enemies along with your teammates and botmates(creeps, minions, droids, they got different names in different MOBAS). Then you level during the progress, you can attack neutral enemies around the map, you can buy items, pick up spawned items etc.
Awesomenauts is a bit different since it's more focused around twitch and platformer skill, you can think of it as a more fast-paced MOBA game. It also is a bit unique in the sense that it's a bit simplified, although not simpler as in easier. It got a bit of different difficult though, it's more tactical while other MOBA's I would say is more strategic.
I consider Awesomenauts not as hardcore or competetive as lets say DOTA, but there is some competetive activity. For example I found this site earlier today:
http://awesomenautsproleague.com/