You'll probably be best off heading to the articles sub-forum for in-depth how-to's, but since you asked, cube is simply a custom draft format. That is, take some cards you already own, shuffle them up, pass out 3 sets of 15 to each player, draft as normal, then play.
Most cubes are aimed at having the most powerful cards available to the owner in the draft, so often different peoples cubes tend to converge and in these forums there lots of discussion about which cards of each color to include.
Some cubes are built around other restrictions or themes, like mimicing a draft format of the past, being multiplayer friendly, or only using common cards. Often cubes are built with highlighting specific synergies in different color combinations, like Azorius blink or golgari reanimator, etc.
Cube is a user designed set of cards used to either draft or play sealed events, but instead of having to buy packs you just build them from the cube and put them back when your done.
Here is a copy of my Pauper (meaning only using commons) cube
Most cubes are aimed at having the most powerful cards available to the owner in the draft, so often different peoples cubes tend to converge and in these forums there lots of discussion about which cards of each color to include.
Some cubes are built around other restrictions or themes, like mimicing a draft format of the past, being multiplayer friendly, or only using common cards. Often cubes are built with highlighting specific synergies in different color combinations, like Azorius blink or golgari reanimator, etc.
Here is a copy of my Pauper (meaning only using commons) cube
http://www.cubetutor.com/viewcube/63745
And here is an article giving a more in depth look at cubes
http://magic.wizards.com/en/articles/archive/how-build/building-your-first-cube-2016-05-19