Barack Obama's Certificate of Live Birth
In 2007, Hawaiian officials sent Obama his proof of birth documentation in the only form they have issued in this century. It is a document called a "Short form birth certificate (SFBC), a "Certificate of Live Birth" or COLB for short.
Posting an image online is the easiest, most efficient way of getting information out to the public these days, so the Obama campaign posted a scan of his COLB online in June 2008. The best pictures are at FactCheck.org. That organization sent a team to photograph the document after Obama, on campaign at the time, invited reporters to come inspect it. You can enlarge the images to see details such as the raised seal, Signature stamp, folds, number, etc. It is a real document issued in 2007. It looks like most birth documents do these days.
Below is a slideshow of the nine photographs of the COLB taken by FactCheck.
The Department of Health in Hawaii has released an index listing for Obama's birth.
Obama II, Barack Hussein
The birth index entry was created when the original birth certificate was finalized. Hawaiian officials have said that they they looked at Obama's birth record and he was born in Hawaii. That, plus the birth document posted online at Factcheck.org, has been enough for everyone but a few conspiracy theorists.
As soon as the COLB was released, bloggers claimed that it was a photoshopped fake. Here's an example of one debunking: analysis of about the COLB being photoshopped.
The Hawaii Dept. of Health maintains a web page devoted to Obama's birth certificate FAQ here.