A Beautiful Mind. The life story of John Forbes Nash Jr., a Nobel Prize winner who suffered from schizophrenia. This is an excellent movie IMO.
Shrek and Shrek 2. Computer animation sendups of popular fairytales. Good jokes if you can recognise the fairytales they send up and good music soundtracks.
The big three from Japanese director Akira Kurosawa, The Seven Samurai, Yojimbo and Sanjuro. These films inspired The Magnificent Seven, A Fist Full of Dollars and For a Few Dollars More (it's ironic that Kurosawa made these movies after watching US westerns and was criticised in Japan because they were "too American", and then these movies inspired more westerns). They are all black and white with Japanese voicetracks (with English subtitles). There is some action in them but the action is not the point of the movies. If you want something a bit more intellectual there is Kurosawa's Ran, based on Shakespear's tragedy King Lear, but set in Feudal Japan and his version of Macbeth, Throne of Blood.
Brassed Off. Set in a declining English coal mining town, the mine is on the verge of closing (it's the town's biggest employer) and the town brass band is struggling to keep going. An "underdogs succeed against the odds" movie, with some implied criticism of the English Thatcher government policies of the 80's which accellerated the mine closures.
The Full Monty. Another English movie in another declining industrial town (Sheffield). A group of unemployed men who struggle to find work amid company closures and downsizing decide to put on a male strip show to bring in some cash. No, it's not ghey, it's a comedy, and a good one at that.
The Italian Job (the 1969 original, not the 2003 remake). A "bank robbery" movie with a very original car chase through Rome.
I don't know how many people would say these movies are "must see/have", Beautiful Mind is probably the best of them IMO, but they're all worth watching at least once.