Nearly all video games (including Minecraft) are driven by one big program loop. Just as every gear in a clock is synchronized with the pendulum, every task involved in advancing a game simulation is synchronized with the game loop. Appropriately, one cycle of the game loop is called a "tick".
Minecraft's game loop runs at a fixed rate of 20 cycles per second, so one tick happens every 1/20th of a second. An in-game day lasts exactly 24000 ticks, or 20 minutes.
On each tick, various aspects of the game advance a little bit: moving objects change position, mobs check their surroundings and update their behavior, health and hunger are affected by the player's circumstances, and much much more.
One thing that does not happen as part of a tick is drawing graphics. Rendering happens in a separate, asynchronous loop. This prevents video performance from affecting game mechanics, and vice-versa.
Block Ticks
On every game tick, 48 block positions are chosen at random from each chunk and any blocks at those positions are given what could be called a "block tick". Most blocks ignore this tick, but some use it to do something spontaneous: plants grow or die, fire burns out, ice melts, leaves decay, farmland becomes hydrated, and so on.
Because block ticks are granted randomly, there is no way to predict when a block will get its next tick. The average time between ticks is 47 seconds. That is, there is a 50% chance that the interval will be shorter than 47 seconds. But it can occasionally be much shorter than 47 seconds, or much longer. For example, there is a 1% chance that the interval will be over five minutes. For the math behind these numbers, see the Wikipedia entry for the Poisson distribution.
Blocks that react to ticks don't necessarily react to every tick. Some wait for multiple ticks to occur and some have only a random chance of reacting. This is how some kinds of blocks can behave slower or more erratically than others.
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A redstone tick describes two game ticks. This creates a 0.1 (2/20) second delay in the signal of a redstone circuit. That is, the signal's time to travel from a location A to location B is increased by 0.1 (1/10) seconds. A tick only pertains to the increase in signal time, thus, a signal's travel time can never be decreased in reference to ticks.
This diagram visualizes the approach explained in method 1.
A single tick can be produced by placing down a block with a redstone torch on one side and input signal from the opposite face to the torch.
A tick can be produced by placing down a redstone repeater which, by default, has a tick of 1 when the two redstone torches are closest together. Right clicking the repeater will increase the tick by magnitude 1, to 0.2 (2/10). A single redstone repeater can be configured to produce a tick of 4 (4 tick) on its highest setting with the two redstone torches farthest apart. File:Repeater tick positions.png
From left to right, 1 tick, 2 tick, 3 tick, 4 tick.