The Infinite Dungeon (ID) is truly infinite for all practical purposes. The level number is limited by some 32 or 64 bit integer value, but who really cares about that? It would take many lifetimes to descend through all its levels, even assuming a PC could survive such a trip, which is impossible. The danger level increases with dungeon level; any level greater than 200 or so becomes ridiculous.
The ID has some unique characteristics: shops, altars and pools are never generated. Every dungeon level is randomly generated, so leaving a level and reentering it results in a completely new dungeon level. There is one special level, the SIL, where MaLaKaI resides.
According to the Mad Minstrel, the ID was originally a monument built by the powers of law. The incursion of Chaos into Ancardia has twisted it into its current condition.
Any non-hostile mobs, including Filk, that you lose in the ID will re-appear later in the wilderness to meet up with you.
Stair-hopping (thanks to gut for this section)
Stair hopping is the process of repeatedly ascending and descending a single staircase in the ID, for the purpose of generating hordes of items in the continuously generated rooms. Items generated on the ID's floors will be of a danger level equal to 1/3 of the ID's dungeon level. That means only DL1 items will be generated on ID levels 1-3. On ID levels 4-6, DL2 items can be generated, and DL3 items can be generated on ID levels 7-10. This applies to dipping items into potions of raw chaos and exchange as well. This does not apply to guaranteed drops, like eternium gear from chaos warriors. Also, it seems that this does not always apply to monster generation either.
Benefits of stair hopping include: An easy way to acquire spellbooks, as they are all DL1, and thus available on ID levels 1-3. An easy way to acquire extra DV/PV points for the early game, as shields, hats, gauntlets, boots, etc... can sometimes have nice stats. An easy source of teleport control (actually the only guaranteed source of TP control in the game), as blink dogs can be found after ID 4 (sometimes earlier). An easy way to aquire potions and scrolls that can save a PC's life.
Conclusive proof that the SIL is not part of the ID is the fact that altars are sometimes generated in the SIL. These can come in very handy since MaLaKaI can be sacrificed.