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To look at - When you look at something (or someone) you are interested in the appearance of that thing or person.

Generally we look at things that are static.

For example:

Look at these photos, they're really good.
I went to the art gallery to look at the exhibition of paintings.
Look at her, dressed up like a dog's dinner. What on earth does she think she looks like?

To watch is a verb.

When you watch someone or something you are interested in what happens.

Generally we watch things that move or change state.

For example:

I watch TV every night.
The security guard watched the shoplifter steal the clock.

!Note - If I say to you "Look at him!" I mean for you to check out his appearance. But, if I say to you "Watch him!" you can take it as a warning.

See also see vs watch.