Many sites post, or allow users to upload, music that has been licensed for re-use under Creative Commons. These sources can be helpful to find music for the background of a film or presentation, or for finding tracks to sample. Some restrictions might apply, so look at the information about individual pieces on these sites to determine what you can use and how.
The following are curated collections--often created by libraries--that offer free access to digitized files, usually based around a common theme. They're a great source for finding music that is familiar and of historic significance to use in lectures and for study.
The following streaming audio services are free (or free up to a point). Some allow users to listen to anything in the catalog while others let users to select a representative song or artist and get recommendations based on similarities. Most will require a log in for at least some features. These services focus on commercial releases, usually with more emphasis on popular titles than on classical titles.