The simple answer is that caffeine stimulates the production of adrenaline, which is a hormone often called "the fight or flight hormone". The way this hormone normally works (without caffeine) is that when many nerves in the body are firing excessively (due to fear, pain, etc.) your pituitary gland releases adrenaline, which causes many changes in the body:
a) your pupils dilate
b) your breathing tubes expand to increase oxygen supply
c) your heart beats faster
d) more blood is supplied to big muscles and blood flow to the skin and stomach is reduced (prevent bleeding and avoid wasting energy on digestion)
e) sugar is released into the bloodstream
f) your muscles tense up to prepare for action.

These changes are designed to help you be alert.

Caffeine causes this release of adrenaline through a "fake-out". Here's how it works:
Normally, when you are lying down to sleep, your brain generates adenosine a neurotransmitter which slows down nerve activity in the brain. Caffeine blocks the nerve cells from detecting or being affected by adenosine. It does this by attaching to the same spot on nerve cells used to detect adenosine (these spots are called adenosine receptors).

Because adenosine is what your brain uses to reduce and stabilize nerve activity, suddenly all of the brain cells become hyperactive. This effect causes the pituitary to release adrenaline (it thinks that all the activity means there is an emergency), which leads to the responses I mentioned above.