No...not named for the cypress tree. The name is derived from the ancient Greek word for 'copper', which was mined on the island. The island is now divided in two, with Greek Cypriots running the south and Turkish Cypriots running the north...and a United Nations operated buffer zone between the north and south. The only country in the world to recognize the Turkish Republic of Northern Cyprus is...you guessed it, Turkey (Türkiey).
Pro Tip: If you fly in / out of Northern Cyprus only go to / from Turkey. Every other country in the world views Northern Turkey as an illegal port of entry.
Nicosia: Capital city. Split by a UN buffer zone between north and south. Generally a cool place on each side.
Kakopetria: A great place to consider if you want to visit wine country, see the UNESCO World Heritage Site painted churches...all within reach of Nicosia.