The tap symbol, as found in ability costs.
In Magic: The Gathering, a card is often represented as having been used when it is tapped, or turned on its side. Tapping is one of the major features of the game, and understanding the nuances around it will greatly determine your ability to navigate and anticipate different situations.
Tapping is a feature of permanents, though not all permanents can or will tap. A tapped permanent, having been used, cannot do some things that an untapped permanent can. However, some cards have ways to work around this, such as Chandra, Fire of Kaladesh's ability to untap--to be turned right-side up--when certain conditions are met. You cannot tap or untap permanents unless you are doing so in accordance with abilities or game rules.
Every turn begins with an untap step, in which all tapped cards are untapped. However, some cards restrict untapping, such as Claustrophobia or Dehydration, effectively crippling cards for the opposition. Once tapped, most cards do not untap until their controller's next untap step. As such, it is crucial to take into consideration which cards to use, and when, and in what circumstances.
Lands[]
Land cards tap for mana, the primary resource of the game. A basic land card, when tapped, adds one colored mana of its color to your mana pool, unless cards are in play that state otherwise. Other nonbasic land cards, such as Foundry of the Consuls, can tap in order to provide other effects.
Lands can be tapped for mana at any time, though many cards and abilities can only be played on your turn. However, it is often wise to leave lands untapped to use during your opponent's turn, in order to counteract abilities found on cards such as Archangel of Tithes or to drop a devastating Calculated Dismissal when your opponent isn't expecting it.
Leaving too many lands untapped can serve as a warning for your opponent, who might suspect an instant or activated ability on your part during their turn.
Creatures[]
Creature cards tap most often in order to participate in combat, though they also can tap in order to pay for activated abilities. Because creatures are affected by summoning sickness, they cannot tap on the turn that they are played--unless they have haste, which overrules this restriction.
A tapped creature cannot block during your opponent's turn. It thus becomes very important to determine which of your creatures should attack and which should remain available to block in order to defend yourself from damage. Some creatures, such as Guardians of Meletis, have defender, which prevents them from attacking; these are often ideal for blocking with. However, some cards are quite capable of making untapped creatures incapable of blocking, such as Kytheon's Irregulars or Suppression Bonds.
Sometimes, the only viable way to have enough untapped creatures to block with is to not attack at all.
Artifacts[]
Artifact cards occasionally carry conditions associated with their tapped or untapped state. Though Magic Duels currently lacks most of these, similar functions are found in Ghirapur Aether Grid or Whirler Rogue, which allow you to tap artifacts in order to trigger abilities or gain bonuses. Artifacts untap during the untap step along with all other permanents, unless a card states otherwise.
Abilities[]
Activated abilities, as mentioned earlier, often carry costs that include tapping, as represented by the "
" symbol in the ability cost. These abilities can usually be used at any time, and can change the terms of the game to a devastating degree; the choice between using and not using an Anointer of Champions' ability can be the difference between a live and dead creature, which can upset the overall control of the board.
Tapping for creature abilities can also expose your board to an enemy attack, so the cost of a change to your overall position needs to be factored into your decision for whether or not the ability should be used. This is less of an issue for lands and artifacts.
Unless needed at a precise moment--as in the previous example--many activated abilities are best used at the end of your opponent's turn. This helps prevent them from being interfered with by your opponent's spells and abilities. Because your untap step immediately follows, it also involves minimal cost to your board control and avoids most dangers for whatever you need to do at that time.