There are a lot of lists of 'must-see' movies out there, including the AFI 100 Years... 100 Movies, Academy Award Best Picture winners, the IMDb top 250, Yahoo! 100 Movies to See Before You Die, and TIME All-Time 100 Movies.
I've seen a lot of those movies (including 63 of the AFI list), but there are a lot of movies that don't necessarily make these lists that I'd like to recommend. So here goes:
Some of my all time favorite movies include The Philadelphia Story, 10 Things I Hate About You
(my favorite Shakespeare adaptation), Love Actually, Seven Brides for Seven Brothers, Arsenic and Old Lace, Star Wars, The Princess Bride and An Affair to Remember. I could watch these over and over.
Movies I think everyone should watch are Guess Who's Coming to Dinner and Mr. Smith Goes to Washington. Really, the issues raised in these films are well-treated and still relevant.
My favorite Hitchcock film is To Catch a Thief, although there are so many good ones, that's a tough choice. Also noteworthy are Notorious, Rear Window, Dial M for Murder, North by Northwest, and The Man Who Knew Too Much (1956). I'm actually a big fan of Hitchcock's films.
And yes, there are a lot of old movies on my lists. You really can't go wrong seeing a movie with Cary Grant, Katharine Hepburn, Audrey Hepburn, James Stewart, Gene Kelly, Grace Kelly.
OK, I realized I forgot to add a movie to my list here: Mamma Mia!