Because the Moon is much smaller than the Earth, its shadow can only cover a small part of the Earth's surface, such that solar eclipses can only be seen from certain locations on the Earth. However, the fact that anywhere on Earth can see a total eclipse (where all of the Sun is covered by the Moon) is a quirk of nature in that although the Moon is 400 times smaller than the Sun, it's also 400 times closer, so it can cover the Sun perfectly. Although these rare events occur somewhere on the Earth about every 18 months, it's likely that it won't happen again in the same spot for another 300 to 400 years.
The Moon's shadow has an umbra (a dark cone-shaped shadow, at the yellow point on the diagram) and a penumbra (the purple shaded area). Someone in the umbra sees the whole of the Sun eclipsed, while someone in the penumbra sees only part of the Sun covered.
These events are called a Total and a Partial solar eclipse respectively. A total solar eclipse only lasts for a few minutes, as the Moon's shadow moves east at over 1700 km/h. As a result, a total eclipse can never last more than 7 min 40 secs, and is often much shorter.
Click here to see an animation of a total and partial eclipse.
To find out where in the world there will be an eclipse over the next 10 years see this page: Solar eclipses.
When viewing an eclipse it's important not to look directly at the Sun, as even when it's mainly covered it is bright enough to cause permanent eye damage, and can even blind you! For tips on how to safely observe an eclipse, the Royal Astronomical Society (RAS) and the Society for Popular Astronomy (SPA) have created a very useful PDF guide that can be downloaded from the following link:
In addition, the SPA have created an 8 minute video for the 2015 event visible from the UK, but which can be applied to any eclipse. This can be viewed below: