I was wondering if I could claim my brothers son on my taxes this year. He does not work and we all live together. Is it possible to claim his son even though he is not mine?

If you meet the requirements, you can claim both your brother and his son.
A dependent can be
a. Your Qualifying Child,
b. Your Qualifying Relative.

Your brother can be your Qualifying Relative if he stayed with you as a member of your family and did not earn $3,400 or more.
Your brother's son can be your qualifying child only if he is not qualifying child of his parents.

Read IRS Publication 17: Your Individual Income Tax; Chapter 3: Exemptions for Dependents.
Internal Revenue Service

Agreed, but no one else cannot claim the child in your place or before you who has an equal claim, like, say, a grandparent.

So you should go through Chapter 3 of publication 17, and then post here if you need any further clarification.

Not a bad idea.