To put it short, Digimon are the name of the creatures that inhabit the Digital World, a parallel dimmension to the ours in the fictional franchise of the same name. They are composed of computer data, just like everything in their world, and come in a variety of forms with different powers and abilities.
When they get stronger, they evolve into more powerful forms and despite being made of data they are able to materialize into the human world just like the humans are able to cross the portal into their dimmension with no ill effects.
When a Digimon is destroyed they are usually reconfigured as an egg, from which they will revive in their weakest form, so their life cycle is basically infinite: they die, reborn, grow stronger and die again. It, however, is not a golden rule, there are ways to kill a Digimon for good and in some continuities, like Tamers, the revival cycle doesn't exist.
They also have their names following the pattern of [x]mon, "x" usually being what they are supposed to represent, but they can also be a pun or similar stuff related with the subject. Leomon, for example, is a lion Digimon. There are some species that don't use the -mon suffix, but they are a rarity.
In most continuities they don't have physical sex differences, being born of accumulated data coming from our world, but they are known to use the psychological concept of gender. Again, not a golden rule, in some continuities Digimon have physical sex differences and they born from a sexual relationship, like Cutemon in Xros Wars.
For more detailed information go to the Digimon Wiki.