Learn German: Mein Bruder ist klein.

Because Bruder (brother) is a masculine noun. When a masculine word is the subject of the sentence (in the Nominative case), the word for "my" is just mein. If it were a feminine word like Schwester (sister), you would use meine.

Why doesn't the word klein get an ending here?

When an adjective comes after a verb like ist (is) to describe the subject, it stays in its basic dictionary form. You only add endings to adjectives when they come directly before the noun, like in the phrase mein kleiner Bruder (my little brother).

Does klein mean "little" as in young, or "short" as in height?

It can mean both! In German, klein is used for things that are small in size, but also for people who are short in height. Depending on the context, this sentence could mean "My brother is little" (younger/small) or "My brother is short".

Why do we use ist instead of bin or bist?