In general, you're only supposed to use royalty-free songs so you don't get sued.
However, if you're a student and it's a project that you do not profit from, you might get away with using a copyrighted song. But I strongly advise you to use royalty-free songs. And always give credit to the artist/songwriter/copyright holder.
Here's a good rule: Will youtube or facebook let you upload your video? If they do, your video should be okay. (Note: I had to type in and prove that I had permission to use the songs on my facebook-approved videos. In fact, you should try and see if facebook will let you upload your videos (which you can delete later). If facebook says it's okay, it should be legal and ready to submit for competition.
Click below for some royalty-free songs (but I'm not 100% sure). These should be okay for school projects where you're not charging money.