There is much disagreement over what specifically defines language. Some scholars define it by six properties: productivity, arbitrariness, duality, discreetness, displacement, and cultural transmission. (I have found some lists of five, but these often combine two of the various six into a single characteristic.)
Productivity means language has some purpose beyond making sounds, gestures, or markings. Language helps us accomplish tasks and share information.
Arbitrariness acknowledges that the words we speak, gestures we make, and markings we use in writing do not really represent what we are talking about. For example, the word "apple" is not really an apple, just a word we have assigned for discussing the fruit. There is an exception to this in speaking, though: onomatopoeic sounds—like crash, bang, and plop—directly mimick an actual sound.
Duality is the property of language that is at work when we create words or gestures which have meaning out of sounds which are otherwise meaningless. An easily recognized instance of duality is the use of prefixes and suffixes. For example, the sound re- on its own does not have any meaning, but when it is added to the beginning of a word in English, it means to do something again, as in revisit, retry, and review.
Discreetness of language is somewhat connected to duality. The many sounds and symbols we use in language (like the sound re-) are understood to be separate entities from one another. We use the letter A when we mean to convey a certain sound, and we don't use other letters in place of it. Different languages may have different sets of sounds or symbols to work with, but it is this set of symbols and sounds which acts like a palette of distinct colors to make words from.
Displacement is at work when we talk about people, things, or ideas which are not presently happening around us. This flexibility is highly distinctive of human language. Most other animals are only capable of creating call sounds in response to present stimuli—food, danger, or a desire to mate, for instance. As humans, we can say things like, "I wonder what I'll have for dinner later," even if we are not presently hungry.
Finally, cultural transmission is a vital part of human language. While other animals are born with an understanding of their bodily and vocal languages, humans learn language through the process of enculturation. This aspect of learning language is of interest to many psychologists, and studying people who have not been exposed to language can provide deeper understanding of human development and the role language plays in our lives.