If you're prone to doing something, it means you're likely to do it, have a habit of doing it, or are susceptible to it. People who are prone to getting the flu every winter should probably get a flu shot in the fall.
The second meaning of prone is "lying face down," and the Latin root is pronare, "to bend forward." This is a handy way to remember both meanings: if you tend to wait to start your term paper until the night before it's due, you are prone to procrastination — you bend toward it. If you stay up so late you fall asleep face down in the book, you are now prone at your desk.