Negima! is a direct adaptation of the original manga. It covers most of the major stories from the manga up through Volume 6, then does an original story for its last few episodes. The stories are done out of order, though; the Library Island arc from Volume 2 is done after the Evangeline arc from Volume 3, and the filler stories are scattered all over the place. The Kyouto story arc is compressed into two episodes (it made up the majority of Volumes 4, 5, and 6 of the manga).
Negima?! is a reimagined version of the story created by Akiyuki Shimbo and Shaft, with the usual weird art style. I like to refer to it as "Pani Poni Negi", because the art and humor are very reminiscent of Shaft's Pani Poni Dash. The story of Negima?! is completely different from the manga, and is more focused on absurd humor and less on action and fanservice. (There is also some action and a decent amount of adventure, but almost no fanservice.) The characters are vaguely similar but also quite different in many ways: Asuna is obsessed with finding a chupacabra, Makie for some reason believes that a frog creature is her father, and Konoka is sarcastic and clever instead of cheerful and ditzy. (But she's sarcastic and clever in a cheerful way.)
The Spring and Summer Specials are OVAs of Negima?! I haven't seen them but according to Wikipedia, the Spring Special is based on Volume 7 of the manga, where Asuna and Negi get in a fight at an island resort, while the Summer Special is an original story about Nodoka.