101 Dalmatians (1961) - Though I like the 1990s live-action remake better (the original book is wonderful!)
Becket (1964) - Haven't seen this in a long time - haven't let the kids watch it yet
The Great Escape (1963) - Not the most well-crafted movie ever, but tells an important true story and I've always liked the music.
Guess Who's Coming to Dinner (1967) - Fascinating exploration of racism through the story of a liberal white couple (Spencer Tracy and Katherine Hepburn) whose principles are challenged when their daughter decides to marry a black man. A little on the mature side as I remember it.
Lilies of the Field (1963) - Haven't seen this in ages, though I remember liking it a lot.
The Magnificent Seven (1960) - Classic Western based on the Japanese movie The Seven Samurai. I believe this had the same director and the same composer as the Great Escape (a number of the same actors as well).
A Man for All Seasons (1966) - A well-loved favorite in our house.
The Miracle Worker (1962) - One of my all-time favorite movies. Intense. Almost allegorical.
To Kill a Mockingbird (1962) - Great movie. Great book. Too intense for the littles.