Companions are very useful allies in Demise when used properly.  They are most useful when playing a single character, though they can be useful when in a party also.
You earn about 25-30% of the experience when companions damage/kill monsters. While this may make companions sound very bad, they are actually fantastic if you use them properly, and when playing a solo character, they are virtually vital.  When solo, companions can take hits for you, allowing you to stay longer in the dungeon.  Companions can kill a monster for you (or at least keep it occupied) if you get paralyzed.  Many companions have special attacks that can kill large numbers of monsters, or instantly kill an enemy.  A companion killing a group of ZBrats, or stoning that nasty beheading creature has obvious benefits...

When working with companions, keep the following points in mind:
1)  Keep party size at 6 or less.  Any more than this and companions tend to hit other party members instead of the monsters you are fighting.  This can be a nasty problem if these companions have special abilities.(Which are the companions you will often aim for)
2)  Keep your companions well healed.  Just like your characters, companions fight better when they have high HP.  This might be slightly less important if you have a companion with a huge amount of HP, but you should aim to keep HP around 3/4.  It is also particularly important if you have charmed a companion that is too large for you to carry!
3)  Keep companion bind level up! For companions that have joined, make sure they are well healed if you want to keep them, else they will get restless very quickly and leave.  With charmed monsters this is also important, you don't want a companion leaving on you, and you definitely don't want it attacking you.
4)  Beware of fright traps! Unless your companion is bound for life, this is an instant companion 'killer'.  This especially sucks if you have just charmed 'big bad monster #53' and were going to life dominate it when you got back to town, but accidentally set off that fright trap...  (AAARRRGGGHHH!!)
5)  Don't die.  While this is generally a given aim for players at any time, it becomes especially important if you have a hard to get companion bound for life - dying will eliminate all that hard work!
6)  Companion order seems to work similar to party order.  The companion in front seems to face most of the attacks, with progressively less being aimed at companion 2, 3 & 4.
 
 
Companions To Aim For:

Spellcasters
Allows you to use them to quickly blast through horde creatures without wasting any of your mana.  If possible, taking a quick trip down a level or two to get a more powerful spell caster is very valuable.  Coming from down deeper, these spellcasters will have higher A/D and HP which will allow them to survive better against more powerful creatures on your current level.

Breathers
Dragons tend to dominate this particular area, though there are other creatures that can breathe fire/cold.  Very helpful for creatures with Complete Spell Res, as breath weapons can do large amounts of damage to large groups.  Dragons also tend to have decent HP & A/D, so are useful even without their breath weapon.

Warriors
They have high HP & A/D - can generally cope with many other monsters on the level, as they can hit well and defend themselves reasonably.  They do tend to lack resistances however.

Instant Killers
Creatures with Stone(1) or Behead(1) are great!  Instantly killing an opponent is one of the greatest gifts for an adventurer.  But if these attacks get misdirected at a companion or a character... well, don't say you weren't warned.  This doesn't tend to happen with a solo character, but if you get four characters joined up with four stoning companions... you're just asking to die. :)
 

Oh, and a final few points.
Give your companion a nice name - after all it is your 'lovely' pet/buddy.(Or perhaps lover if your character is overly sick & twisted)
Mistreat weak companions that offer to join, showing your 'true' companion how much it means to you.
Take it to the companion store occasionally, so it can socialize with other friends with similiar mental ability.(Or lack thereof)
Give it milk & cookies every now & then. For undead try intestine soup, and dragons tend to like live goblins or slaves...