Ahhh the old "what's your favorite dive site" question...
You'll get as many answers as there are sites.
The truth is, if you have never been to Coz, they're all good. I like to say, "stand on the water's edge and point out to sea...that's the good diving on Coz."
But just to get things started, here's my top list:
Shallow dive: Columbia Shallows, you'll need a shovel to go deeper than 30', but by far this is an all-time top Cozumel dive site for many, many divers. Lots and lots of fish and bottom time to match.
Mid-reef and deep dives:
1.) Columbia Bricks is by far one of my favorite deep dives, it has it all, swim-thrus, pinacles and walls.
2.) Francesa & Yucab for mid-reef garden dives.
3.) Tormentos, always a crowd pleaser, beautiful reef dive, turtles, ocotpus, morey eels, baracuda all regulars on this dive.
4.) Palancar Gardens, lots of deep sea fans, large grouper and lots of blue chromis milling about. (actually any of the 4 Palancars are great Gardens, Deep, Caves or Horseshoe...love horseshoe)
5.) Villablanca Wall, a great true wall dive, just a sheet steep slope covered with barrel sponges and lots of sea life.
This is a VERY short list, and changes depending on my mood :d I personally pefer to talk to the Dive Master and see where he thinks we should go.
Those guys dive the reefs every day and know better where to take you to see things you might want to see, rather than just tossing out a divesite name because someone says it's thier favorite site.
On all deep dives you can expect to see, lots of blue chromis, midnight parrot fish, other types of parrots; French, Queen and Gray Angel fish, different types of butterflies, tangs, and grouper.
Baracuda are readily available, hawksbill sea turtles, Eagle rays, Southern stingrays, electric rays and the ever present Yellow stingray all can be seen.
As for sharks, the most common shark is the nurse shark, but black tips are occaisonally spotted at the southern dive sites.
Night dive: Paradise reef is a very nice dive during the day, my second favorite shallow dive, but at night it is very special with the night critters running about.
Also Villablanca Shallows, the area between Blue Angel Resort and Hotel Villa Blanca, is a nice daytime shore dive, with lots of juvinile critters to explore, but a great night shore dive, especially around the Sting ray enclosure right next to Blue Angel, just hang out there and see what shows up.