The United States Supreme Court, in a unanimous decision written by Justice Thomas, found that you pretty much cannot patent genes. More specifically, merely separating out naturally occurring DNA is not patentable, but synthetically made (i.e. complementary DNA or cDNA) could be patented. This makes sense, as there has been a long standing rule against patents on naturally occurring things, laws of nature, and natural phenomenon.
The ruling is summarized by Justice Thomas as follows: 
[W]e hold that a naturally occurring DNA segment is a product of nature and not patent eligible merely because it has been isolated, but that cDNA is patent eligible because it is not naturally occurring.
The patent at issue involved genes (called BRCA1 and BRCA2) isolated by Myriad Genetics. The genes were involving mutations which indicate a high likelihood of developing breast cancer. Myriad Genetis apparently conducted significant research and developed an expensive set of tests to find those mutations and argued that others companies testing for the same breast cancer genes would violate Myriad Genetics’ patents. However, the Supreme Court rejected their arguments, noting that [g]roundbreaking, innovative, or even brilliant discovery does not by itself satisfy the §101 inquiry [for whether something can be patented]. Discoveries are generally not inventions, and cannot be patented.
The Supreme Court wrote:
It is undisputed that Myriad did not create or alter any of the genetic information encoded in the BRCA1 and BRCA2 genes. The location and order of the nucleotides existed in nature before Myriad found them. Nor did Myriad create or alter the genetic structure of DNA. Instead, Myriads principal contribution was uncovering the precise location and genetic sequence of the BRCA1 and BRCA2 genes within chromosomes 17 and 13. The question is whether this renders the genes patentable.
This landmark case will invalidate a large number of gene patents, especially considering that the US Patent Office has routinely granted gene patents for the last several years. Many, like Myriad, believe that such a ruling stymies research and may destroy the biotechnology fields. After all, a good chunk of ones patent portfolio may now be considered void. On the other hand, some companies may be able to do more exploration on genes without fear of patent infringement suits.