The island was mostly used for military purposes in the 18th and 19th centuries - and still houses many former barracks - but it was fully converted into a public park in 2010.
Bike paths have been built and it hosts a myriad of food events, concerts and exhibits throughout the Summer - you can see the full schedule here. Free ferries leave for the island every hour during the week and every 30 minutes at the weekends and it takes less than 10 minutes to get there.
I first went there for Meatopia last year and returned last Sunday when the Boy and I went to a BBQ, Blues and Beer festival there for the afternoon. You can't really go wrong with these three things put together and we spent hours nibbling on pulled pork, mini burgers and hot dogs while chilling out and enjoying the laid back atmosphere.
I highly recommend visiting Governors Island as a cheap and very easy way to get that "out of the City" feeling if only for a day!