I'm a Buddhist. What does this mean?

To me this means that I go for Refuge to the Three Jewels. The Buddha, his Teachings, and those who follow his teachings, are the most important things in my life. In practice it means trying to be ethical in my actions, speech, and thought. It also means that I do meditation and other spiritual practices - such as puja.

But what is Buddhism? Many more learned and wiser individuals than me have summed up the Buddhadharma but here's my attempt.

The Buddha taught that all phenomena are dependent on conditions, that everything is in a constant state of flux. He also taught that there are only phenomena, there is no numena - no underlying essence, no permanent entity, no soul. Because we do not see phenomena in this way, we attribute solidity and permanance to them. This attachment causes us to be disatisfied by our experience of the world ie we suffer. The Buddha also teaches that everything we do, say or think has consequences. By acting in our ignorance we set up the conditions for continued suffering and, according to the tradition, for continual birth, death and rebirth. The fact that suffering is ubiquitous should tell us that something is wrong. Pain in the body is a sign of illhealth. For the Buddhist "suffering is important not for its own sake, but only because it is a sign that we are not living as we ought to live." (Sangharakshita). Buddhism aims to awaken us to the true nature of phenomena, and to this end devised many many methods to lead us to that awakening - meditation being the method par excellence.

I practice with the FWBO and my Teacher is Urgyen Sangharakshita

I'm interested in the connection between Buddhism and the martial arts.

Buddhist Literature.
While it is a good thing to read books about Buddhism I believe that it is important to read Buddhist scripture. We are blessed to be able to read a large number of Buddhist scripture in English translation these days. In fact many sources are now available on the web. See here for references to readings I like.