|MadSci Network: Astronomy|
Satellites like Skylab fall back to Earth due to the effect of atmospheric drag. In low Earth orbit (like the majority of missions and all manned missions since Apollo), there is still some drag from the outermost regions of the Earth's atmosphere. This problem doesn't affect the Earth orbiting the Sun---we are well clear of the Sun's atmosphere---and therefore we can be reasonably confident our orbit is stable on timescales of many millions of years.
[Making this more definite, how far away from the Sun's atmosphere is the Earth? That depends partially on how far out one considers the Sun's atmosphere to extend. The outer reaches of the Sun's atmosphere is called the corona. From the image in the link, one can see that the corona extends a few times the Sun's radius from the Sun itself. Let's assume that most of the Sun's atmosphere is within 10 solar radii of the Sun. By comparison the Earth's orbit is about 200 solar radii in size. Thus, the Earth's orbit is well outside of the Sun's atmosphere. For further reference, the Moon orbits about 50 Earth radii away. Thus, in a relative sense, the Earth is farther from the Sun than the Moon is from the Earth. Given that the Moon is still orbiting quite nicely, we should have no worries of plunging into the Sun.
In the distant future, some 5 billion years hence, the Sun will expand into a red giant. Its size will increase to roughly the size of the current orbit of Venus or Earth. However, at the same time, the Sun will also lose a considerable amount of its mass, which means the sizes of all of the orbits of the planets will increase. We do not understand the later portions of a star's life well enough to say with much certainty what will happen, but the current thinking is that the Earth's orbit will expand just enough that it will not plunge into the Sun. Needless to say, of course, it will still be close enough to the Sun that the Earth will not be a pleasant place to be! Moderator]
Try the links in the MadSci Library for more information on Astronomy.