Evolutionary changes occur for a variety of reasons, and the "five fingers of evolution" is a useful memory tool to help a person remember five common causes of evolutionary change within a population or species. It's helpful because it unsurprisingly uses each of the fingers on the human hand.
Before discussing each "finger," let's clarify the type of evolution we are talking about. For the purpose of this answer, evolution is a change in the gene pool of a population over a period of time, and the five fingers of evolution explains five ways to change that gene pool.
The first finger is the little finger (pinky). It is smallest finger, so this reminds us that populations can shrink. Populations shrink for all kinds of reasons. Natural disasters, war, epidemics, are more are all causes of population loss, and the individuals lost are likely determined completely by chance. Those that survive are the new gene pool, and the distribution of specific genes is likely no longer exactly the same as what it was. A change in the gene pool (evolution) has occurred.
The next finger is the ring finger. A ring finger usually wears a wedding ring, and this ties in to being a part of a couple. Genetically, this should remind us of a concept called non-random mating. For example, if a given population simply stopped mating with any individuals that have a particular trait, then that trait will eventually be eliminated from the gene pool.
The third finger is the middle finger. This is the mutation finger. "M" from the word "middle" is the memory trigger for "mutation" because they both start with the same letter. New genes and genetic coding can enter into and change the gene pool through mutation.
The fourth finger is the pointer (index) finger. When you want someone to move, you tend to point. This finger reminds us of gene flow. As individuals immigrate and emigrate, they change the frequency of the genes in a particular gene pool.
The final finger is the thumb, and it refers to natural selection. Natural selection is dependent on adaptations increasing or decreasing fitness. Organisms with adaptations that increase overall fitness are likely to survive. Those adaptations are essentially being given a "thumbs up." Adaptations that are not good for an environment are given a thumbs down and are not likely to survive long enough to pass on the trait. That in turn changes the gene pool and propels evolution forward.