Actually, we can go right to the source on this: George Gaylord Simpson's Tempo and Mode in Evolution (1944).
Tempo is the rate of evolution. Simpson was a paleontologist, so he was trying to answer questions like "what is the percent change in the size of this tooth per million years?" He has a chapter estimating these rates for various characters in horses, which have a very good North American fossil record.
Mode encompasses the mechanisms responsible for evolution and their patterns. Simpson was working at the time of the Modern Synthesis (the wikipedia page is decent and brief enough on the topic, about which books are written), coming from the perspective of a paleontologist trying to reconcile genetic evolution with the macroevolutionary patterns he was seeing in the fossil record.
Simpson was really the first paleontologist to look at fossils in a quantitative way, trying to understand of connections between microevolutionary processes and macroevolutionary patterns. He even co-authored one of the first quantitative and statistical methods books for "organismal" biologists: Quantitative Zoology.
Specifically, of tempo, he says (p. xvii):
"...evolutionary rates under natural conditions, their acceleration and deceration, the conditions of exceptionally slow or rapid evolutions, and phenomena suggestive of inertia and momentum."
Of mode, he says (p. xviii)
"...the study of the way, manner, or pattern of evolution, a study in which tempo is a basic factor, but which embraces considerably more than tempo. The purpose is to determine how populations became genetically and morphologically differentiated, to see how they passed from one way of living to another or failed to do so, to examine the figurative outline of the stream of life and the circumstances surrounding each characteristic element in that pattern."
Mode basically encompasses all the processes of evolution. Tempo is the subset of mode that concerns how fast evolution is (or is not) happening.
The great thing is that 70 years after Simpson's book, biologists are still trying to understand tempo and mode of evolution.