Learn French: Il n'y a pas de montre que j'aime. - There is no watch that I like.

We use que (that) because it represents the direct object of the verb. You are the one doing the liking (the subject), and the watch is what you like (the object). Whenever the noun being replaced is receiving the action in the new clause, you must use que. If the watch were doing an action, you would use qui.

Can I skip que and just say Il n'y a pas de montre j'aime, like in English?

No! In English, we often drop the word "that" and simply say "There is no watch I like." However, in French, you can never drop the relative pronoun. You must always use que (or qu' before a vowel) to connect the two ideas.

Why is it pas de montre instead of pas une montre?

In French, when you make a sentence negative using ne ... pas, the indefinite articles un, une, and des (a, an, some) change to de. So, il y a une montre (there is a watch) becomes il n'y a pas de montre (there is no watch).

How exactly does Il n'y a pas break down to mean "There is no"?