I divided the important episodes into 2 categories. Those that influenced Trek in general - spilling over into the movies and spinoffs, and those that defined the characters the most.
Star Trek in general definers:
1. Errand of Mercy - defines the relationship between Feds & Kilingons
2. Balance of Terror - same as above for Romulans and adds defense to Enterprise's mission.
3. Journey to Babel - Brings diplomacy & politics into Trek, adds Spock's parents, Andorians and Tellarite's, introduces "needs of the many outweigh needs of the one" philosophy.
4. Mirror, Mirror - added a whole other "alternative universe" idea to Trek that repeats in spin-offs, and novels. This new movie borrows the idea.
5. Enterprise Incident - added that the Federation as not always perfect, and can do some questionable things.
Character definers
1. The Naked Time
2. Shore Leave
3. This Side of Paradise
4. Trouble with Tribbles (defined the minor characters a little)
5. Amok Time
6. The Savage Curtain (I hate this one but it does define)
7. Tholian Web
8. Court Martial (I hate the Kirk deification but it does define)