Learn French: Il y a trois bouteilles d'huile dans cette boîte. - There are three bottles of oil in this box.

You are right that 'some oil' is usually de l'huile. However, after a specific unit of quantity like bouteilles (bottles), French drops the article and just uses de (which becomes d' before a vowel). This is the same rule you use for combien de (how much/many).

Does il y a mean 'there is' or 'there are'?

It means both! Unlike English, which switches between 'there is' for singular and 'there are' for plural, French uses il y a for everything. Here, it translates to 'there are' because it introduces a plural quantity (trois bouteilles).

Are the 's' endings on trois and bouteilles pronounced?

No, both are silent. The 's' at the end of French plural nouns like bouteilles is almost never pronounced. The 's' in trois is also silent here because the next word (bouteilles) starts with a consonant.

Why do we use cette instead of ce for 'this'?

Cette is the feminine form of 'this' or 'that'. Because boîte (box) is a feminine noun, we must use cette. If the box were a masculine object, like a bag (sac), you would use ce sac instead.