Things that are contemporary are either happening at the same time or happening now. Contemporary art is recent art.
In history class, if you hear that one famous person was a contemporary of another, that means they lived at the same time. Contemporaries are people and things from the same time period. Contemporary can also describe things happening now or recently. It's common to speak of contemporary music or contemporary furniture, for example. Those things are new, not old. Anything characteristic of the present day can be called contemporary.