Evidence for plate movement can take many forms. You can select topics from the adjacent icons. Before the model of plate tectonics was properly formulated, its precursor theory - continental drift - was based on the jigsaw-like reconstruction of continents. Long term relative motion between plates can be inferred by matching geology between continents that can thus be deduced to have once been connected. This approach was used by geologists in the early 20th century. Quantifying the motion of continents relative to the Earth's magnetic field, also over the long-term, can be done using palaeomagnetic methods.
An alternative reference frame for charting plate motion comes from hot-spots. These are deduced to remain nearly stationary (at least moving much more slowly than the plates) in the asthenosphere and produce volcanic islands (in the oceans, just volcanic masses on the continents) at the surface. So as plates move the islands more chains. However, the really key information for determining the rates of plate motion come from the magnetic anomaly patterns of the ocean crust (yet to be added to this site).
To find the long-term (averaged over several million years) rates of relative plate motion - click here. That plates are moving today can be demonstrated from earthquakes. The sense of relative movement of the earth on either side of seismically active faults can be determined from focal mechanisms - any for big-shallow earthquakes, can be directly measured from ground motion. But slow steady motion (of a few cm a year) has required improvements in surveying - achieved through space science. These include external reference frames (satellites, distant stars) and hence chart absolute motions.