As others have said, you should do both. Here's why:
You want to validate input on the client side first because you can give better feedback to the average user. For example, if they enter an invalid email address and move to the next field, you can show an error message immediately. That way the user can correct every field before they submit the form.
If you only validate on the server, they have to submit the form, get an error message, and try to hunt down the problem.
(This pain can be eased by having the server re-render the form with the user's original input filled in, but client-side validation is still faster.)
curl or from a script, for example?
(This is not theoretical; eg, I worked on a travel search engine that re-submitted the user's search to many partner airlines, bus companies, etc, by sending
POST requests as if the user had filled each company's search form, then gathered and sorted all the results. Those companies' form JS was never executed, and it was crucial for us that they provide error messages in the returned HTML. Of course, an API would have been nice, but this was what we had to do.)
Not allowing for that is not only naive from a security standpoint, but also non-standard: a client should be allowed to send HTTP by whatever means they wish, and you should respond correctly. That includes validation.
Addendum - December 2016
There are some validations that can't even be properly done in server-side application code, and are utterly impossible in client-side code, because they depend on the current state of the database. For example, "nobody else has registered that username", or "the blog post you're commenting on still exists", or "no existing reservation overlaps the dates you requested", or "your account balance still has enough to cover that purchase." Only the database can reliably validate data which depends on related data. Developers regularly screw this up, but PostgreSQL provides some good solutions.
Yes, client side validation can be totally bypassed, always. You need to do both, client side to provide a better user experience, and server side to be sure that the input you get is actually validated and not just supposedly validated by the client.
I am just going to repeat it, because it is quite important:
Always validate on the server