This is a great picture showing what wingtip vortices and how they move.
When taking off as, this Airbus A300 is, the turbulence will spread out and move away from the airplane along the ground at about 2 to 3 knots until their energy is dissipated. If you are in a small airplane on the taxi way this can really ruin your day. There have been many instances of unaware pilots being flipped over.
When landing the vortex tends to sink below and behind the large aircraft. So you have to be very careful if you are cleared, or are in line, to land behind the large aircraft. You have to keep your altitude higher than the large aircraft did when passing the same point in space. As the vortices are only created when the wings are producing lift, your touchdown point should be further down the runway than the large aircraft.
There was a recent report of a man flying his brand new Cirrus SR22 Turbo behind a Boeing 747-200. He was not paying attention to his position relative to the 747, and strayed in to the 747's vortices. His aircraft was torn apart. Not a fun way to die.