To learn to recognize meter, remember that (in most Western music) the beats and the subdivisions of beats are all equal and even. So you are basically listening for a running, even pulse underlying the rhythms of the music. For example, if it makes sense to count along with the music "ONE-and-Two-and-ONE-and-Two-and" (with all the syllables very evenly spaced) then you probably have a simple duple meter. But if it's more comfortable to count "ONE-and-a-Two-anda-ONE-and-a-Two-and-a", it's probably compound duple meter. (Make sure numbers always come on a pulse, and "one" always on the strongest pulse.)
This may take some practice if you're not used to it, but it can be useful practice for anyone who is learning about music. To help you get started, the figure below sums up the most-used meters.
To help give you an idea of what each meter should feel like, here are some animations (with sound) of duple simple, duple compound, triple simple, triple compound, quadruple simple, and quadruple compound meters. You may also want to listen to some examples of music that is in simple duple, simple triple, simple quadruple35, compound duple, and compound triple meters.