Learn French: Ne regardons pas ce film. - Let's not watch this movie.

In French, you create a "let's..." command by using the nous (we) form of the verb but completely dropping the subject pronoun. Because there is no nous in front of it, regardons changes from a regular statement ("we watch") into a suggestion or command ("let's watch").

Where do ne and pas go when making a command negative?

You wrap ne and pas around the verb, exactly like you do in a normal sentence! Since the verb here is regardons, the negative command simply becomes ne regardons pas (let's not watch).

Why does the tu form of regarder drop its 's' in commands, but regardons keeps its 's' here?

The rule about dropping the final 's' in the imperative mood only applies to the tu form of -er verbs (so "watch!" to a single friend is just regarde). The nous and vous command forms keep their exact present tense spelling, which is why regardons still ends in 's'.

Why use ce film instead of just le film?