The only thing you need to protect your eyes from is the Sun. A lunar eclipse, no matter what phase it is in, is not dangerous to look at. A partial or an annular solar eclipse are dangerous because the Sun is still visible. A total solar eclipse is perfectly safe to look at—however, as the phenomenon starts and ends with a partial eclipse (as the Moon slowly blocks the view, then slowly moves away), those parts of the (otherwise) total solar eclipse are dangerous to look at directly.
DO NOT LOOK DIRECTLY at the Sun with the naked eye, even though some people have done it for VERY SHORT times (seconds, at most) without permanent damage. Again, DON’T DO THIS! The problem is the eye doesn’t have pain receptors, so you wouldn’t know when you had been looking at the Sun for long enough to be hurting your eyes. This can range from a small “blind spot” in your field of view, that goes away after a few days or weeks, to complete, permanent blindness.
There are safe ways to view the Sun through an optical instrument (which, because it collects more light than the eye, would be even MORE dangerous to look through without protection!), and some of them are really affordable financially. The best, however, is a solar filter that goes in front of the objective, NOT at the eyepiece (as the temperature there is so high, it could melt and make you permanently blind INSTANTLY)! Such filters come in many different qualities and many different prices, but are generally rather affordable as well. They are made by applying a special metallic coating to an optical-grade piece of glass or even plastic (Mylar).
DO NOT LOOK AT THE SUN THROUGH AN EXPOSED PHOTOGRAPHIC EMULSION (yes, those still exist!) and DO NOT LOOK AT THE SUN THROUGH A POLARIZED FILTER (unless you’re also using a safe filter as described above).