Palmyra is in Syria.
Palmyra (the roman name) is known as Tadmor to the Syrians. Both mean the same thing - date palm. The name comes from the lush oasis adjacent to the city which is home to some million date palms. It is the only oasis in Syria.
The major tourist attraction of the area is the stunning ruins - the most famous and well-preserved of which are the Temple of Bel, the colonnade, the funerary towers, the hypogeum of 3 brothers, and the Arab castle. All are within a few kilometers of each other.
There are no ATMs in Palmyra, and the local exchange office will not change traveller's cheques. Make sure you have sufficient cash for your time in Palmyra.
If hiring a private car, you might want to consider side trips to Qasr al-Heir ash-Sharki - a partially excavated Ummayad palace quite literally in the middle of nowhere - and Rasafa, originally a Roman city with heavy Byzantine influence, also used by the Ummayads before being destroyed in the Abbasid era. Rasafa is also of interest for the stone it's built out of, more a quartz-like crystal instead of the usual granite or sandstone. Makes for a unique appearance. This route takes you quite close to the Euphrates, and you can be dropped off in Raqqa, Aleppo, or Hama. Car hire can be price, and the driver still has to get back to Palmyra. Expect to pay as much as S£5000 (US$100), if you're heading for Aleppo or Hama. A bit less to Raqa.