The short answer: not much.
OK, let's give credit where credit is due. Genes all have regulatory sequences, which are short part of the DNA code, usually situated slightly upstream from the coding region, where regulatory proteins can bind. For instance, a transcription factor may bind to one and thereby increase the rate of the gene's transcription. So, even though regulatory sequences aren't transcribed, they still have an important function. However, they are usually short sequences only, and don't add much to our figure of 1.5%. At most, the coding regions that play a significant role in crafting the human form account for some 5%.
That's still terrible. What about the rest? Well, as I've mentioned, the rest is mostly "junk DNA". Its constituents include, but are not limited to:
1. Repeat elements
These include tandem repeats and interspersed repeats. Examples of the latter include long interspersed elements (LINE) and short interspersed elemtents (SINE) genes. PZ Myers describes them so perfectly that I'm simply going to resort to quoting him at length here:
LINE, itself a gene about 6,000 base pairs long, codes for an enzyme called a reverse transcriptase. The key here is that it recognizes its own RNA sequence, and repeatedly inserts copies of itself into our genome. LINEs seem to be relics of the "copying machinery" of old viral infections wherein a virus would embed a portion of itself into our genome, not enough to propagate the full, infectious virus, but enough to continue copying itself. It is a classic example of a selfish gene: It has no purpose but to do only that, without benefit to us. Another repeated element in the genome is a shorter sequence called a SINE, or Short Interspersed Element. These are only a few hundred base pairs long and don't actually do anything, as they don't code for a functional protein. They do contain regulatory elements that trigger the cellular machinery to make RNA from them, however, and this SINE RNA has a selfishly advantageous property: It is recognized by the LINE reverse transcriptase, which can obligingly insert duplicate SINEs back into the genome. There are overwhelming numbers of these repeated elements in the human genome: about a half-million copies of LINEs and about a million copies of SINEs, taking up about 45 percent of the total DNA.
These are mobile genetic elements that have the ability to move from one part of the genome to another. When they move, some also leave a copy of themselves behind. Transposons are best seen as parasitic DNA, selfishly going along for the ride, and are seldom of benefit to the host (i.e. us!). These transposons account for about 10% of the human genome.
Pseudogenes are old copies of functional genes that have fallen into disuse. They are not transcribed, but they do look a bit their functional cousins.
These aren't usually included in the definition of "junk DNA", but seeing as the majority of most introns don't seem to have any phenotypic effect, I like to classify them thus.
So, most of our genome seems to serve no purpose. This may seem shocking, but when you think about it, it's almost inevitable. First and foremost, evolution is blind - there's no oversight from some intelligent being capable of removing the inefficiencies detailed above. Secondly, DNA is copied trillions of times in each of us. It stands to reason that it will sometimes make mistakes. Thirdly, the zeal with which DNA is replicated can be 'exploited' by nucleic acids that abuse this process to replicate themselves.
Lastly, there is much talk about possible roles for much of this non-coding DNA. My own take on it all is this: it is almost inevitable that functions will be established for some of it. However I predict that only a very small portion of it will ever be convincingly shown to be of any benefit to us. Large portions of the non-coding 'deserts' within our genomes will remain best seen as "junk". In this regard, there is one trap to avoid falling into. Just because a function can sometimes be found for a portion of our DNA, this does not mean that this function is the reason for that DNA's existence. For instance, LINEs may inadvertently help create new genes, but if they do, then this is only a happy byproduct of their existence. The reason they exist is that they are good at getting replicated. Nothing more.