What happens to the ejection seat after a pilot ejects?
It depends on the seat. Most modern seats, including the ACES II system common in US Fighter Jets, fall away from the pilot at a pre-determined altitude. [src]


  Once out of the plane, a drogue gun in the seat fires a metal slug that pulls a small parachute, called a drogue parachute, out of the top of the chair. This slows the person's rate of descent and stabilizes the seat's altitude and trajectory. After a specified amount of time, an altitude sensor causes the drogue parachute to pull the main parachute from the pilot's chute pack. At this point, a seat-man-separator motor fires and the seat falls away from the crewmember. The person then falls back to Earth as with any parachute landing.


Generally, according to that link, this occurs at just over 100-200 feet above ejection height. However, if you eject at high altitude, such as Captain Scott O'Grady did (26,000 ft), the seat will fall away on a sensor (either 10,000 or 18,000 feet, I don't recall which). The chute may open automatically, or if it is not set to at that height, you can then pull the chute manually, or wait until its set height to open.