In MtG, what are counters and the different types of Counters used for?

Answer 1

Depending on the usage and slang, counters can mean two different things in M:tG.

One form of "Counters" is a short form of "Counterspells" These spells are instants that prevent another spell from occurring as it is being cast. For example Remove Soul has "Counter target creature spell" which means that you would cast it in response to an opponent casting a creature spell, and it would cause the spell to be sent to the graveyard instead of entering play.

The more common usage of "Counter" in M:tG is for markers that are put on spells in play. There are a wide variety of counters from plus counters for creatures (e.g. +1/+1, +1/+0, etc.) to charge counters for artifacts, to all kinds of various counters dictated by cards. Generally, these counters that go onto spells in play remain there until the spell is disposed of (creature killed, artifact destroyed, enchantment disenchanted, etc.) but some cards use the counters for different effects, like removing a counter from Sun Droplet to gain one life at the beginning of a turn.

The editors over at Wizards wrote some short articles about counters when Power Conduit was released in the Mirrodin set, as it allowed counters to be moved among spells in play, converting different types of counters into +1/+1 counters for creatures, and charge counters for artifacts.
http://www.wizards.com/default.asp?x=mtgcom/daily/mg90
http://www.wizards.com/default.asp?x=mtgcom/arcana/445

Interesting tidbit -- for a time, the old Alpha/Beta/Unlimited card Time Vault was re-worded in the oracle text requiring that it's controller sacrifice a turn to put a "time" counter on it, and this "time" counter was sacrificed on a later turn to get its benefit (this was to get around the fairly obvious problem that there would be ways to untap the Time Vault without using a turn to do it, such as with Twiddle, or a Galvanic Key) This idea has since been scrapped, and the current Oracle wording requires a player to choose at a time the Vault would become untapped, either to untap it and sacrifice a turn, or to not untap it.

Answer 2

There are several types of counters. You have Blue counter spells that keep people from playing anything. You have spells that add 1/1 counters to creatures to make them stronger. You have counters that you can remove and get a creature or deal damage. Just go to www.magicthegathering.com and go under the gather and look up counters.