What is negative space? In art it is the space around and between the subject of the artwork. It's an intricate part of composition. Salvador Dali, Egon Schiele, David Downton, Saul Bass were masters in using this to their advantage. Here's an old example on a book cover for The Great Gatsby. Gatsby is illustrated sitting in a cushioned chair, however the chair can't be seen. His body creates the shape of the chair. This is negative space.
Lately we've seen quite a few logos pop up that use negative space in clever ways to create another object, letter or image. This creates intrigue, wit and a uniqueness to the idea, as the viewer discovers the "surprise" inside the mark. This is a great technique to create more of a self-contained object or letter. You can house an initial cap or object inside that creats a concept to help depict the company's personality.
Creating simple logo forms is the hardest thing to do. Here are some great examples of negative space logos.