To philosophize is to think philosophically or just deeply and reflectively. On a long car trip, after you run out of school gossip, you and your friends might philosophize on the nature of man, or the question "What is beauty?"
To philosophize is not exactly the same thing as doing philosophy. Often if we say someone is philosophizing, we are kind of making fun of him, as if all of the sudden he thinks he’s Socrates, but really he’s just parroting Dr. Phil. If you stop driving in the middle of the road to tell your friend a revelation you’ve just had about the nature of suburban society, she might say, “Stop philosophizing and drive!”