In general, the "tick rate" refers to how many times per second essential game tasks are run - this is a "game tick". Examples of tasks which might run on the game tick include calculating the movement of players, recalculating lighting and processing AI behavior. Many different games have different tick rates by default - Minecraft, for example, tries to run at a fixed tick rate of 20 ticks per second, while Source engine games can vary wildly. A higher tick speed doesn't necessarily mean that the server costs more (although it can, depending on the how resource-intensive the game is).
Since the tick speed affects how frequently different tasks can be run, a lower tick rate may make the game feel less smooth while often ensuring that the game doesn't slow down when processor-intensive tasks are run and seem to "lag", while a higher tick rate can make a game feel smoother while taking the risk that a processor-heavy task might slow down the game tick and create the feeling of lagginess.
(As a side note, graphics and networking aren't usually affected by the tick rate - generally only game logic is.)