A collection of nodes or values called vertices that might be related; relations between vertices are called edges.
Many things in life can be represented by graphs; for example a social network can be represented by a graph whose vertices are users and whose edges are friendships between the users. Similarly, a city map can be represented by a graph whose vertices are locations in the city and whose edges are roads between the locations.
Simply put, a cycle occurs in a graph when three or more vertices in the graph are connected so as to form a closed loop.
Note that the definition of a graph cycle is sometimes broadened to include cycles of length two or one; in the context of coding interviews, when dealing with questions that involve graph cycles, it's important to clarify what exactly constitutues a cycle.
A graph whose edges are directed, meaning that they can only be traversed in one direction, which is specified.
For example, a graph of ariports and flights would likely be directed, since a flight specifically goes from one airport to another (i.e., it has a direction), without necessarily implying the presence of a flight in the opposite direction.