Counters are objects you place on cards in order to keep track of things. For example, cards with suspend use time counters in order to show when they will be played. Counters can be anything you have handy, such as small beads, buttons, cereal pieces, anything you want as long as it stays clear exactly how many counters are on the card. Dice can be useful for this, as well.
Counters can also modify power and toughness, but currently +1/+1 counters are almost used exclusively to avoid confusing the players. However, -1/-1 counters have found new use in the Shadowmoor block. A state-based effect provides that if there are both +1/+1 and -1/-1 counters on the same card, they are removed in pairs until only one type remains.
Power and Toughness Modifying Counters
Power and toughness modifying counters often come in the form of +1/+1 counters and -1/-1 counters, although they are certainly not restricted to such. (+X/+Y counters will be used as examples here; X and Y may each be any integral value) They add to or subtract from the power and toughness of a creature. A +X/+Y counter adds X to a creature's power and Y to a creature's toughness.
+1/+1 counters are the most commonly used type of Counter used in the Keyword mechanics Devour, Bloodthirst, (Scavenge) , Graft, Modular, Sunburst, Kicker (some cards), Reinforce, Amplify, Clockwork cards, Spike creatures, Vampires and Hydras and as a main Theme in Lorwyn and Morningtide.
Although first seen in the first sets with cumulative upkeep and rarely used since, these counters were brought back in Shadowmoor and Eventide as a main theme and in the keyword mechanics Wither, Persist. In the Scars of Mirrodin Block, a new ability called Infect uses -1/-1 counters as well.
Other power-toughness modifying counters
Counters modifying the power and toughness of creatures in other ways besides +1/+1 and -1/-1 have been avoided since shortly after the beginning of the game but are still in use. Among them are -0/-1 counters only seen in the Ice Age Block, -2/-1, +2/-1, -2/-2, +2/+2, +1/+0 and +0/+1 counters.
Seen on enchantment cards, especially in Zendikar and Worldwake. Such cards are Expeditions and Quests. Examples are Soul Stair Expedition, Quest for the Gemblades, and Zektar Shrine Expedition. With these cards, quest counters are built up by certain events, such as landfall or when a creature deals combat damage to another creature. When such a thing happens, the Expedition or Quest card gets a quest counter as designated by the card text. When the card has a certain number of counters, you may activate the ability of the card.
Poison counters were introduced in Legends, and are important to players in a new way of losing and winning the game. In Future Sight, poison counters were keyworded into Poisonous, and in Scars of Mirrodin they were keyworded into Infect. If a player has ten or more poison counters, they lose the game.
Other Named counters
Other named counters have been used widely throughout Magic: The Gathering usually to either thematically unify cards or to provide additional flavor to the cards. Among them are Age counters for Fading, Ki and devotion counters in Kamigawa Block, Age Counters for Cumulative Upkeep, Flood counters which signify turning lands into Islands and many more with many different purposes.