The first thing to remember about observing an eclipse is safety. A lunar eclipse — an eclipse of the Moon — is perfectly safe to watch with the naked eye; you're only looking at the Moon, at night, which is quite safe. A solar eclipse is potentially dangerous, however, because viewing a solar eclipse involves looking at the Sun, which can damage your eyesight.
This page therefore contains some information on eye safety during a solar eclipse.
A solar eclipse can be viewed safely with the naked eye only during the few brief seconds or minutes of a total solar eclipse, when the Sun itself is completely obscured by the Moon. Partial eclipses, annular eclipses, and the partial phases of total solar eclipses are never safe to watch without taking special precautions.
Even when 99% of the Sun's surface is obscured during the partial phases of a total eclipse, or in an annular eclipse, the remaining sliver of the Sun is intensely bright — just as intense as at any other time — and cannot be viewed safely without proper eye protection. Failure to use appropriate filtration may result in permanent eye damage or blindness.
Looking at the Sun through any kind of optical aid (binoculars, a telescope, or even a camera's viewfinder) is extremely dangerous, and can cause permanent blindness.
There is no pain or discomfort when the retina is being burned, and the resulting visual symptoms do not occur until at least several hours after the injury has occurred; by which time it is far too late.
To look at the Sun directly, you must use proper solar viewing protection, such as eclipse glasses. Sunglasses do not provide anything like adequate protection, as they do not block the wavelengths of light which are likely to damage your eyes, or reduce the intensity of the visible light sufficiently. Since much of the damage is done by invisible infra-red light, the fact that the Sun appears dark in a filter or that you feel no discomfort does not guarantee that your eyes are safe. In fact, sunglasses can make it worse: as they block visible light, the pupil in your eye widens, letting in more harmful UV and infrared light.
Properly designed solar filters, made and certified to appropriate national safety standards, are therefore the recommended protection for direct viewing. These glasses should be much darker than regular sunglasses; they need to filter out ultra-violet, infra-red, and 99.997% of the visible light. They should be reasonably new, and in good condition. If in doubt, or if they appear to be damaged at all, destroy them.
Various other ad-hoc solar filters are sometimes discussed; but in practice these can be dangerous, and so can't be recommended. In any case purpose-designed eclipse viewing glasses are readily and cheaply available, so it's simplest and safest to get the real thing.
Fake goods of all kinds are becoming increasingly common; unfortunately, this can apply to solar viewing glasses too. So be sure that you obtain whatever viewing aids you use from a reputable source.
Viewing the Sun indirectly, by projecting its image onto a screen, is a safe way to enjoy any solar eclipse. You can make a projector with a simple pinhole, or with binoculars or a telescope, as described in Observing Eclipses. However, never look through the projector — only look at the image on the screen.
Note that a screen refers to a matte surface, such as a white sheet, or a piece of paper, so that the Sun's image can be seen by anyone looking at it from any angle. Looking at a reflection of the Sun in any shiny surface is basically the same as looking directly at the Sun, and as dangerous.
In spite of these precautions, the total phase of a solar eclipse — when the Sun is completely covered by the Moon — can and should be viewed without any filters. The naked eye view of totality is safe and is the most awe-inspiring astronomical phenomenon you are likely to see. Just remember to look away and put your eclipse glasses back on as soon as the Sun returns.
There's been such hysteria — allbeit well-intentioned — stirred up about eye damage, that many people are convinced that it is specifically solar eclipses that cause eye damage; that is, at any other time of year, looking at the Sun is OK. This is not true: looking at the Sun at any time for more than a second or two can cause permanent eye damage.
Finally, I've heard some truly daft ideas for eclipse viewing, such as looking through a sheet of Perspex, or in a reflection in a bucket of water. I have no idea where these come from, but these are not safe! If you can see the Sun clearly and brightly, whether directly, in a reflection, or via Perspex, then it's dangerous.
For more information by real experts, I highly recommend that you read these guides: