I'm currently searching for songs which have older songs in it part-wise. I don't even know what this term is called. I'm thinking of something like Eminem's Sing for the Moment which contains parts of Aerosmith's Dream On.
The term you're looking for is probably sampling. To quote from Wikipedia, sampling is
the reuse of a portion or sample of a sound recording in another recording. Samples may comprise rhythm, melody, speech, or other sounds.
As for examples, you can try searching on Google using keywords like 'list of sampled songs' or the like. There are a lot of these songs and it would require an extremely long list if I put it here.
Edit: As mentioned by @Bebs, you can also try checking out whosampled. This lists songs and the samples used or if they were sampled or covered in other songs.