Rom hacks are alterations of original games and can alter anything from sprites, levels to asm changes which effect how the game plays. Contrary to popular belief, big companies also do rom hacks. Super Mario Bros 2, the US version is a rom hack of Doki Doki Panic with sprites and some subtle assembly alterations
Super Mario Bros 3, being as popular as it is, has quite a few different hacks floating around for it.
Level hacks are among the most common hacks as they require the least skill to perform with the help of editors such as Mario Improvement 3. Some hacks will also change maps and the behavior of objects on maps.
Depending on the degree of skill by the level editor, these can look like they were made in blender, as good as the original or something revitalizing to the game such as Super Mario Bros 3: Chaos Control.
Another almost as common, and even more common in the case of some games, is Sprite hacks, these change various parts of how the game will look. It can be as simple as the main sprite for you character being changed to redesigning the whole background. Music hacks are also sometimes included which can be either the movement of a few mids to different places in the game to all out rearangements of the scores in some cases.
Often hacks like these are used to enhance level hacks although they are quite common just for the case of creating offensive and stupid works of art.
The type of hack has to deal with changing the behaviors of the game via changing the code. This can be essentially used to change the behavior of the game or use the engine of one game to create something completely different. This type of hack is often used with other hacks. It can be used to change the behavior of simply how one computer controlled object behaves or how fire balls leap to major changes such as changing p switches to effect the timer or making coins be used to buy objects in mushroom houses or many other possibilities. In the case of Super Mario World, such hacks have been made to completely redesign the game.
I'll classify this hack as language as these hacks were originally used for said purpose. A hacker takes the original text and replaces it with new lines to say a new message. While said hacks are often used to translate a game into another language, such hacks can also be used to change stories, often in a more crude manner.