"Metroidvania" is a term used to describe a sort of exploratory 2D platforming game. It comes from the original games of the Metroid series and some of the middle-era of Castlevania games (most notably "Symphony of the Night", which is what primarily gave birth to the term).
The most defining feature is usually the world. Rather than have set stages or a world map, the entire game takes place inside one giant map, which you must traverse about. Not just any map, but typically a very complex map filled with all kinds of secrets. The map is typically divided into sectors, more in an aesthetic fashion to denote that the denizens and dangers will be different. Sometimes you have teleports to assist you in travel, but other times you just have to remember where to go in the map.
There's various elements at play in the exploration. There's typically tidbits hidden in spots, or puzzles that have to be returned to when you find appropriate equipment. The exploration is also non-linear - your limits of progression through the map are by various tools or powerups that you need to acquire, but the order in which you must acquire some of these tools is not set. And in many choices, there are many paths to get those tools, or even get past obstacles without those tools.
A very popular element of these games, especially in the Metroid franchise, is the concept of "sequence breaking". Using special tricks or otherwise mastering the utility of your basic abilities, you can sometimes access items and areas far earlier than the game expects you to.
As far as I know, Metroidvania games have always been 2D platforming games. Aside from the elements of exploration, the actual gameplay of Metroidvania games vary widely.
The genre has a few popular names, including:
- Hack and slash
- Action RPG
- Dungeon crawl
- Diablo-like or Diablo-esque
And sometimes even the term "roguelike" is used - there is a vast difference between the roguelike genre and games like Diablo, but the latter are obviously influenced by the former.
The most common term I've encountered is "hack and slash". There are many, many games that belong to this genre, especially since the success of Diablo.
The excellent Moby Games site has a page about the genre, which briefly outlines what defines it as well as providing a list of all the games it considers to be "Diablo variants".
Best Answer
There is not a clear definition of MOBA out there, so let's define it.
MOBA is a genre definition for multi-player games where not only two parties struggle for resources or attack/defend, but each party/person is against all others in a confined space, therefore the term
arena.Battle arenas in real terms had a pre-defined resource collection: Multiple entry points for opponents, and spread out in the arena, weapons and shields.
Loosely defined, one can see it as a mixture of RTS/RPG and Free-for-all deathmatch, the emphasis is in some games on the former part (DoTA-type games) and in others on the latter (Monday Night Combat).