David Bowman is a character in a story jointly written by Stanley Kubrick and Arthur C. Clarke, called "2001 A Space Odyssey" (which is both a book and a movie). The character later appears in the sequel to book 2010 Odyssey Two and the sequel to the movie 2010 The Year We Make Contact. The character also returns in 2 more books by Arthur C. Clarke, 2061 Odyssey 3, and 3001 Final Odyssey.

Warning: Wikipedia contains spoilers

In the movie, and in its sequel (2010 The Year We Make Contact), the character is played by Keir Dullea.

David Bowman in 2001: A Space Odyssey
In both the book and movie 2001, astronaut and scientist David Bowman is on a mission to track the source of an alien artifact found on the moon, which leads to a moon around the planet Jupiter (in the novel, Saturn); the story takes place in the year 2001. He has to interact with both a human crewmate and an intelligent supercomputer named HAL 9000. The HAL 9000 computer utility suffers an apparent malfunction which leads to the death of all humans on board except David Bowman. Through determination and cunning, David is able to outwit the supercomputer and regain control of the spaceship Discovery. Upon orbital insertion around Jupiter, however, David Bowman is confronted with a huge, black featureless monolith with the exact proportions of 1 by 4 by 9 (the squares of the first 3 ordinals). He takes a space pod to investigate, and he is greeted by arguably the most surreal sequence of scenes in science fiction film history.

David Bowman in 2010: Odyssey Two (book) & 2010: The Year We Make Contact (film)
In the end of the 2001 movie, David Bowman is led through a metamorphosis from a material, soul-bearing being to an energy-like being. This state, and David Bowman's incarnation of himself in this state, is referred to in the 2010 book as the Star Child.
The visual representation of David Bowman's new state in the movie(s) is that of a variety of human David Bowman bodies at various ages (from embryo to elderly).
In the movie 2010, David Bowman attempts to communicate with the main character by proxying messages through HAL's communication system. The other character at first assumes that HAL or human crew member is playing a trick. Ultimately, David shows himself to the character.
Toward the end of the 2010 story, David returns to Earth, briefly meeting with his wife and then his mother. In the book, David as the Star Child defuses an orbiting nuclear weapon, derailing the onset of a war.
(continuation of short description of David Bowman's actions in the book/movie 2010 need to go here)

David Bowman in other books by Clarke
{short description of David Bowman's role in Clarke's other books in the series -- Sorry I haven't read them yet)

Etymological significance of David Bowman's name
The first name "David" is believed to point to the biblical David, who killed the much stronger Goliath through skill not power. The second name "Bowman" is assumed to be a reference to Ulysses, whose skill with the bow is often described in the Odyssey. Ulysses also blinded the one-eyed Polyphemus with a trick, as Bowman deactivates one (camera) eyed HAL 9000 is quite clear. Hence one reason for the name of the film: "2001 A Space Odyssey".