Learn Turkish: Bu bir kalem.

In Turkish, there is no separate word for "is" in simple sentences like this. The meaning is understood just by putting the subject (bu) and the noun phrase (bir kalem) together.

Does bir mean "one" or "a"?

It acts as both! Bir is the number one, but it also works as the English words "a" or "an". In this sentence, it translates to "a" because we are simply identifying what the object is, rather than emphasizing that there is exactly one of it.

Since the current lesson is about ne (what), how would I ask "What is this?" to get this exact answer?

You would ask Bu ne? (This what?). In Turkish, you often answer a question by simply replacing the question word with the answer. So, you drop ne and put bir kalem in its place to get Bu bir kalem.

Could I leave out bir and just say Bu kalem?

You could, but it changes the meaning slightly. As a full sentence, Bu kalem usually translates to "This is the pen" (a specific one). Adding bir makes it clear you are just identifying it as "a pen" in general.