The original conjectures concerning plate tectonics were based on circumstantial evidence like the shapes of continents being such that they would fit well if pushed together. Today, we have a much broader set of evidence in favor of the hypothesis.

Indications of Tectonic Activity

Among the classes of evidence for continental drift and the underlying plate tectonics we may list

1. The shapes of many continents are such that they look like they are separated pieces of a jig-saw puzzle. For example, look in the adjacent map at the shape of the east coast of North and South Americal relative to the shape of the west coast of Africa and Europe.

2. Many fossil comparisons along the edges of continents that look like they fit together suggest species similarities that would only make sense if the two continents were joined at some point in the past.

3. There is a large amount of seismic, volcanic, and geothermal activity along the conjectured plate boundaries. This is shown clearly below in the figure labeled "Crustal plate boundaries" where the epicenters of earthquakes above Richter magnitude 5.0 are plotted for a 10-year period. The concentration is striking, and indeed this plot serves to define the plate boundaries extremely well. 
4. There are ridges, such as the Mid-Atlantic Ridge (see figures above and below) where plates are separating that are produced by lava welling up from between the plates as they pull apart. Likewise, there are mountain ranges being formed where plates are pushing against each other (e.g., the Himalayas, which are still growing).

Crustal plate boundaries

Age of the Sea Floor

If the crustal plates are pulling apart at boundaries like the Mid-Atlantic Ridge (see the line of earthquake epicenters down the center of the Atlantic in the preceding figure), the sea floor near these ridges should be very young geologically, since it is formed of material upwelling from the interior. This is indeed the case, as the following figure shows.

Age of the sea floor crustal plates

This figure displays the estimated age of sea floor crustal plates with red the youngest and blue the oldest . Click on the image for a larger one with legible writing (but it is a 385 kB file). One can see clearly that material near the crustal boundaries is very young geologically.

Information supplied by: http://csep10.phys.utk.edu