Learn Turkish: Abim benden büyük. - My older brother is older than me.

In Turkish, comparisons are made using the ablative suffix -den or -dan (which usually means "from"). To say someone is older "than" you, you literally say they are older "from" you (benden büyük).

Doesn't büyük usually mean "big"?

Yes, büyük primarily means "big" or "large". However, when talking about people's ages, Turkish uses büyük to mean "older" and küçük (small) to mean "younger".

Is abi the only word for "brother"?

Abi specifically means "older brother". If you want to talk about a younger brother, you use erkek kardeş. You might also see abi spelled more traditionally as ağabey, but abi is how it's usually spoken and written in everyday life.

Where is the word for "is" in this sentence?

In Turkish, the "to be" verb for the third person ("is") is often invisible. You could add the suffix -dir to make it büyüktür for a factual or formal tone, but in everyday speech, just ending with büyük perfectly means "is older".