Learn Turkish: Şu bir masa. - That is a table.

In Turkish, simple sentences about what something is do not need a verb for "is" when talking about a third person or object like this or that. You just put the subject (Şu) and the thing it is (bir masa) next to each other.

How would I ask "What is that?" to get this exact answer?

You would ask Şu ne?. Notice how the answer keeps the exact same word order, just replacing the question word ne (what) with the answer bir masa (a table).

What is the exact difference between şu and o if they can both mean "that"?

We use bu for things very close to us. We use şu for things a little further away, or when we are specifically pointing something out to the listener for the first time. We use o for things that are very far away or completely out of sight.

Can I skip the word bir and just say Şu masa?

Yes, but the meaning changes. Şu masa usually acts as a single block meaning "That table" (for example, "That table is big"). By keeping bir, Şu bir masa clearly forms a full sentence meaning "That is a table."