Broadly speaking, "communication" can be defined as any action taken by an animal (or plant, or other organism) that alters the behavior of another animal without using physical force. Communication happens when a sender passes information to a receiver, through a physical medium that can distort or degrade the message, or contain competing messages.
There are three broad categories of communication. Signals are where the sender actively tries to pass a specific message to specific receivers. Cues, such as the bright coloration of monarch butterflies that tell bird predators that they are poisonous and taste bad, are always "on" and take no effort on the part of the sender to send. Signs, such as footprints, aren't intended to communicate anything but nevertheless provide information that another animal can act upon.
Animals communicate for a variety of reasons. Some examples are to attract mates, threaten rivals, warn about danger, or share information about food sources. The different ways in which animals communicate can be visual, audial, or chemical (smell or taste). For highly social species, there may be facial expressions, body language, or dances.