Briefly, evolution is defined as "the change in allele frequency in a population over time", where:
Allele frequency = the percentage of a population having a specific allele (form of gene)
Population = a number of individuals of the same species (sometimes the phrase "who interbreed" may be added)
So, in plain English, evolution occurs when the kind or frequency of traits in a number of organisms changes over a period of time.
Now, to answer your question: evolution is not entirely random. While the appearance of mutations may be random, the process of selection is not random. Rather, individuals who survive to reproduce pass on their genes; if a particular trait is more beneficial in this regard, a higher percentage of individuals with this allele will reproduce that those without the allele. This will eventually lead to a change in allele frequency over a period of several generations (evolution).
Evolution also occurs through genetic drift, and the trait distribution within a species may be influenced by population bottlenecks, either through natural disasters or through founder effects (ie. a pregnant rat surviving a shipwreck gives birth on an island devoid of rats; the genes of the mother and her mate will strongly influence the traits of future generations). These are what can be said to be essentially random events of evolution.
For speciation to occur, evolution must normally be coupled with some sort of reproductive or behavioral isolation. I can explain that if you like, but it's not entirely relevant to your question.