Alfred Wegener brought together several lines of evidence to support his theory of continental drift. One is quite simple -- that the continents look like they could "fit" together, much like puzzle pieces that have drifted apart. Then, he noticed that when you put the continental puzzle pieces back together, other things started to fit. For example, the rock layers that form the Appalachian mountains of the eastern U.S. matched quite well with those in Scotland. Fossils found on the east coast of Brazil match quite nicely with fossils found in western South Africa. Also, he noticed that a lot of the fossilized life found in the rock record didn't fit in the climates they were found in. For example, rocks in Alaska contain fossil palm tree leaves, though there have not been palm trees at that latitude for a very long time! Thus, he concluded that the continents must 'drift' around the Earth, occasionally colliding with one another. http://scienceline.ucsb.edu/getkey.php?key=2842

Sea Floor Magnetism
Stripes of magnetic material in the seafloor provide strong evidence for tectonic theory. The stripes alternate between those with magnetic material orientated toward magnetic north, and those oriented in the opposite direction. Seafloor spreading is the mechanism behind this phenomenon. As new magma forces its way up to the surface, magnetized minerals in the liquid rock orient along the Earth’s magnetic field and then harden as the lava cools. As spreading continues, the material moves away from the spreading zone as if on a conveyor belt. The Earth’s magnetic field flips every few hundred thousand years, and the stripes on the ocean floor show a record of those changes. By estimating when the flips occurred and pairing that with the distance the strips have moved from the spreading zone, scientists can estimate how fast the continents are moving.

Fossil Evidence 
The continents have moved a great deal in the history of the planet, but they carry records of where they’ve been. Some of this evidence is the fossils of animals and plants. Tropical species found in the Antarctic and similar fossils found in western Africa and eastern South America tell a story of where those land masses used to be. Paleomagnetic evidence is an even stronger piece of evidence. Magnetic strata within the fossil record show how the land masses were oriented at different times during Earth’s history. By constructing detailed records of changes in land mass orientation, scientists can reconstruct paths of tectonic movement much further back in history than they can from the magnetic striping on the sea floor.

Direct Measurement
Modern technology gives us a range of ways to directly measure the movement of tectonic plates. These methods are based around the idea of measuring distance between two points on Earth by using some intermediary transmitter in space. For example, SLR (Satellite Laser Ranging) uses two lasers on Earth, each of which fires a laser to a satellite orbiting the planet. The beam is reflected back to each laser and the difference in laser beam travel time is used to estimate the distance between the two lasers on Earth. Another approach using radio-telescopes measuring natural radio signals from deep space called VLBI (Very Long Baseline Interferometry) achieves a similar effect. Both SLR and VLBI allow for repeated measurements of distance between two points. As the continents move, the distances change. Using these measurements, scientists can accurately estimate movement of the tectonic plates today.