Q: "How many different counters have been used in Magic so far and which is the most common (I'd assume +1/+1)?"
A: From Devin Low, Research & Development:
"As you have guessed, +1/+1 counters are by far the most common. In recent years, we've followed an unwritten rule that the only size-changing counters are +1/+1 counters, so you won't see any more wacky -2/-1 or +0/+X counters. This centralization is useful so that these days, when you see a 0/0 with 3 counters on it, you can be pretty sure it's a 3/3. I can remember casting Frankenstein's Monster with a couple of +2/+0 counters, a couple of +1/+1 counters, and one +0/+2 counter, and I think you can see why the new system is much better. Withering Hex was a big temptation to use -1/-1 counters, but the editors stayed the course and worked around it with named Plague counters.
"We also use specially named counters, but these have been centralized too. In all of Onslaught block, Gold, Plague, Trap and Charge may be the only ones. Mirrodin's named counters are intentionally unified as Charge counters, with the necessary exceptions of Fate and Flood. Unifying the charge counters opens the door to cool cards like Power Conduit that manipulate all the counters at once, and turn one card's counters into another's.
"In Magic's earlier days, these unwritten guidelines about counters didn't exist. Designers and players alike could have a field day with different named counters, and the only real rule was: The weirder the name, the better. Some counters that went nameless on the original cards have even been given special names in their official Oracle text. How many different named counters have there been? Are you sure you want to know this? Really?? Well alright, I'll name as many as I can remember, but I can't help it if the answer makes you pull out some old card boxes. I mean who could forget classics like Velocity, Carrion, or Wage counters? Spore, Tide, and Poison counters were crucial to their races' very existence. Magnet, Pain, Arrow, and Paralyzation counters created a lot of 'Magic: The Puzzling' scenarios in actual games. Page, Pin, Infection, and Hunger counters built up on the same card over time, but you usually only put one Bounty, Javelin, or Hourglass counter on a single card.
"Like many of you, I can remember winning and losing games with each of Soot, Verse, and Fuse counters. Fading, Delay, and Arrowhead counters smashed a warpath across limited, casual, and constructed tables alike. Age, Elixir, Credit, Pupa, Currency, and Doom counters build up power over time. Omen and Sleep counters wind down. And Dream, Polyp, and Energy counters yoyo up and down in all sorts of crazy ways. I give the award for most varied use of a single named counter to Growth counters: on Momentum, they're basically glorified +1/+1 counters, while other Growth counters do something very non-basic.
"Counters help a lot to stagger the mana production of non-basic lands, whether releasing it in bursts with Depletion counters, building it up over time with Storage counters, or just balancing a benefit like the beloved Mining counters. Time counters were the first counters not printed on the card, but named in Oracle. The card received errata early on to fix a broken loophole: the unbeatable Elder Druid/Time Vault deck! (Well . . . it sure seemed unbeatable back then.) I just spent 4 seconds trying to think of a combo between Sleight and Petal counters, but it's just not there. My Merfolk Assassin / War Barge deck used to keep guys tapped with Net counters, and I've regenerated a handful of times with Corpse counters. I've never added an Incubation counter, but I've gotten smashed multiple times by the Palladia-Mors that hatched out of them. And if I had to choose between 40 Echo counters and 40 Winch counters…. I know which one I'd pick.
"Well that wraps up the counters, though I'm sure there are several I missed. May you never face an Ebon Praetor with 3 counters on it and have no idea what size it is!"