I could cite many, many examples, but here are three that stand out:
Eva Cassidy's version of "Over the Rainbow." A sublime performance of a classic song by an unbelievable singer. This one always gives me shivers.
"Can't Hardly Wait" by the Replacements. One of Paul Westerberg's best songs, wonderfully recorded by the great Jim Dickenson.
"Backstreets" by Bruce Springsteen. Incredibly dramatic and dynamic track, with a tour-de-force vocal performance. This is the song that really got me to start LISTENING to music -- noticing how a song was put together, arrangements, instrumentation, harmonies, dynamics, etc. They don't call him the Boss for nothing.