The US Navy has the frigate Constitution, launched in 1797. The British Royal Navy has the Victory, which dates even further back -- to 1765. But both of these vessels are museum ships, rather than truly active vessels.
The oldest naval vessel in active service is the VMF Kommuna, a Russian Navy salvage ship built in 1915. James Dunnigan writes for Strategy Page:
This 2,500 ton catamaran was built in the Netherlands and entered service in 1915. Kommuna began service in the Czar's navy, spent most of its career in the Soviet (communist) Navy, and now serves in the fleet of a democratic Russia. Originally designed to recover submarines that had sunk in shallow coastal waters, Kommuna remains in service to handle smaller submersibles, does it well and has been maintained over the decades to the point where it cheaper to keep the old girl operational, than to try and design and build a replacement.
http://www.strategypage.com/dls/articles/Why-A-Czarist-Warship-Remains-In-Service-12-30-2009.asp via Hell in a Handbasket | More Pictures | Image: Warfare.ru