Actually, the sun does orbit things. For one thing, the sun orbits about the center of mass of the solar system (which I believe is still inside sun, as it turns out, but not at the center of the sun) The sun also orbits about the center of mass of our galaxy. Our galaxy orbits around the center of mass of our local group of galaxies and so on...

Actually, this is probably two questions!A) Why does the Sun appear
not to move from the center of the solar system; and B) Does the Sun
orbit anything?

Answer A:

Just looking at the solar system, you might think that the Sun
doesn't move, and all the planets revolve around it. But another way
to think about it is that each body in the system (including the Sun
and each planet) has mass, gravitationally attracting each of the
other bodies; sort of like each one pulling on ropes between them.
From this point of view, the Sun is not stationary, but is moving in
a very small orbit of its own.

To simplify the problem, think of two people sitting on a teeter-
totter, one very large person and one very small person. Where do
each of them have to sit in order for the teeter-totter to balance?
Which one sits closer to the fulcrum(the pivot that you teeter on)? If you try the experiment you will begin to have your answer! Will it balance if one of the people
sits exactly at the fulcrum? This would be like the Sun standing
still at the center of the solar system.

From this you may be able to understand that the Sun does not sit
still in the center of the solar system, but revolves around a point
called the barycenter (center of mass) of the solar system. The
barycenter is the balance point for all of the masses together (Sun
and planets and asteroids and comets and...) in the solar system.
Since each planet is pulling on the Sun with a different amount of
force and with a different orbital period, the Sun's movement is not
a clean circle around the barycenter. It tends to wobble a bit, but
most of the movement corresponds to the pull of the larger planets,
Jupiter and Saturn.

Answer B:

Does the Sun orbit around anything else?

Think about the fact that the Sun (along with the solar system) is
part of a much larger collection of stars (with their planets) called
the Galaxy. Each star system has mass, and they are all distributed
in space to form a disk. If they didn't balance each other's
gravitational pull, they would fall in towards the center of the
galaxy, and make one big sphere of matter (before shortly becoming a
black hole). The reason the star systems don't fall together is that
they are each in an orbit around the barycenter of the galaxy, each both pulling and being pulled by all the other star systems. So, the Sun does have an orbit within the galaxy. Now, can you use this line of thinking to understand what might be happening in clusters of galaxies? Do galaxies have orbits?