Booting means "starting" a computer. The word "boot" comes from the notion of "bootstrapping" which is a reference to a method of pulling on old fashioned high leather boots with straps that were at the sides. Various legends talk of people who could "pull themselves up by their bootstraps". In a similar way, a computer starts with just a few very simple instructions, which call a few more complex instructions, which in turn call even more fancy instructions until the whole computer is up and running.

See the Wikipedia entry on bootstrapping.