There are several ways to look at a solar eclipse without burning out your eyes.

Glasses

Of course, you can buy solar eclipse glasses that make you look like a total nerd.

These glasses filter out harmful UV and IR rays, and the vast majority of visible light, allowing about one part per million through.

Pinhole Projector

You can take an index card and poke a small hole in it. Then place that card a few inches from a flat surface, and you will see an image of the Sun.

If you're in a pinch, arguably the easiest is to make a small hole with your fingers. It doesn't even have to be circular, just small enough.

The bigger the hole, the farther away you will need to situate it from the surface onto which you project the image. The pinhole projector was a few inches from the index card, but my hand was a couple feet away from the wall.

Interestingly enough, a hole of any shape will resolve to the shape of the light source if it is far enough away from the surface onto which it is casting a shadow. NASA shows how you can make some neat patterns with a strainer:

Camera

Sure, you could by an expensive lens for your DSLR camera, or you could place your solar eclipse glasses in front of your camera, but an even easier way to photograph the eclipse is to exploit lens flare in your phone's camera.

This happens because the Sun is really bright and causes reflections inside the lens of your camera. These reflections are significantly weaker than the image of the Sun itself, so the image sensor in your camera isn't saturated there.

Just be sure that you don't destroy your lens by not pointing your camera at the sun for very long! Your lens can start to melt in harsh sunlight.

Or you could get lucky and take a picture with just the right amount of cloud cover like Steve: