The “V” in AV (Audio Video) is one cable and carries only SD (480i) signals.

The 3-plug component video can carry HD (1080p) signals.

Here’s a page that spells it out simply:
The Complete Guide to TV Ports: How to Hook Up Your TV

AV in is your normal Audio/Visual input connection for TV, DVD, VCR, DVR, Blu-Ray, etc.
COMPONENT IN is usually your DVD, Blu-ray etc. connection. The difference, really, is probably the DEFAULT connection (A/V) that the TV goes to automatically.
Older DVD players that didn’t have the HDML cable used a three way cable system for high definition, (which was an upgrade from the three cable RCA connection that was red, right channel sound, white, left channel sound, and yellow, which was for VIDEO).. The cables were color coded, blue, green, red. Usually for the COMPONENT.

@Call_Me_Jay @kritiper Thanks, guys. My (old) DVD player has both AV and COMPONENT outputs and my TV has both AV and COMPONENT IN. Does that mean I can use either one? Which is better?
Also, I think that one port (or plug, whatever you call it) seems to be common to both AV and COMPONENT. How come?

The COMPONENT connection will give the utmost in picture quality and sound.
The cable that seems common to both may actually be a different size. Just use the three cable COMPONENT cables.