The difference between an American citizen sponsoring spouse and children, and bringing a brother or sister to the United States as a legal permanent resident is that in the former situation, there is no wait time. Depending on the nationality of the sibling, the wait time can be years. So the process is actually a very slow one. So this is what you do. Get hold of the USCIS Form I-130 and fill it out.
Obviously, you need to prove that you are an American citizen (a birth certificate for native borns and a certificate of naturalization for those naturalized) and other than that prove that you two are related as siblings (if you are biologically related then just the birth certificates of both of you will do the trick but if there are complications like step parents and/or adoption, you will need to use a combination of marriage and birth certificates to build a case that you two are siblings).
With these documents and a check, mail the packet to USCIS and wait for it to be approved. On approval, the Government mails a packet with bunch of forms to both you and your sibling. Fill up those forms, mail them out, go for an interview at the US embassy, get approved, and then simply catch a flight to America. Your sibling is now a green card holder.