Due to its remarkable biodiversity, with over a third of the local plant species found nowhere else, Socotra has been designated a UNESCO World Heritage Site. With over 40,000 inhabitants, though, it's not just a nature reserve.
Yemenia flies once a week from Sanaa and Aden to Socotra Island Airport (IATA: SCT) near Hadiboh. Felix Airways flies several times each week from Riyan (IATA: RIYT). See http://www.felixairways.com/schedule.php.
There is no public transport, but car rental is available.