is it not individuals mutate and populations evolve?
There are random "errors" in dna replication at conception. These mutations may be carried on into later generations. In most cases they offer no advantage to the organism but at the same time do no damage. If there are changes in the environment there is a remote chance that the mutation may provide a small advantage in coping with the changes. Individuals carrying the mutation have a marginally better chance of surviving than those which do not carry it.
There will be further random mutations in later generations providing some individuals with further advantage ... and so on.
Eventually, only those individuals carrying the advantageous mutations will continue breeding and those without will fail to thrive. Eventually only those with advantageous mutations will remain. The species will have evolved. Survival of the best equipped to survive - the fittest.
No doubt some boffins will fill us in on the significance of genotype and phenotype. What you've said is more or less my understanding of the matter. However, you've also implied that
mutation is the instigator of significant evolutionary change - but that not all mutations are advantageous. I don't know how anyone could actually pinpoint somewhere in the sequence and say "this is evolution, and that is not" - we can only say that certain lines didn't thrive, and there may be billions of those unknown to us.
(As a footnote, The Burgess Shale fossils seem to indicate that there were more distinct phyla in the natural world in the very distant past than there are now - I've always found that odd. Maybe someone could comment, even if it is off-topic.)