that kind of depends on what level of explnation you want, but um...
evolution is change over time in a population, via descent with modification. An example would be that the golden staph bugs in your hospital go from being killed off by methicillin, to being resistant to it, over the time it takes for an outbreak of golden staph infections to happen in the hospital.
It's pretty obvious that this kind of descent with modification happens - look at bugs/drugs, dog breeding, pigeon breeding, agriculture (try getting a loaf of bread out of the original form of wheat...).
How that modification occurs, is where natural selection comes in. Darwin and Wallace reckoned that in any given population, you will have variation. So for example, some of the golden staph bugs living on the taps/door handles/ bed rails in your hospital ward will be resistant to methicillin, some won't. This resistance comes from accidents in transcribing DNA, or damage to DNA that is not subsequently repaired (for example UV light causes DNA to break). All of the bugs, resistant and susceptible, end up in (probably several) patients. The ones susceptible to methicillin get killed off, but the ones risistant to it survive. in this context, those ones are the "fittest" (ie the "best") and they survive to breed, and pass on their characteristics.
So in this case the selection was for methicillin-resistance.
DNA is pretty simple stuff - it's very easy to make mistakes when it's being copied (when you're about to make a new cell), or to break it and mis-repair it. the key to understanding just how things change, is to remember the enormous timescales involved. Animals go back at least 550 million years, probably (depending on which evidence you believe) a billion years. One other thing that is important is that it's not a simply linear set of steps from DNA -> protein -> big organs. There are regulatory genes that switch whole sets of others on and off at different times in your development - so for example if you lost PAX-6 you would still have all the proteins that make up eye tissue, but you wouldn't make an eye out of them.
The issue of the fossil record having lots of gaps is that the fossil record IS just that... stuff getting preserved in rock, and then getting dug up again... it's vanishingly unlikely that any given individual worm from 525 million years ago will be preserved in nice soft mud, not eaten, not squished by an elephant standing in the mud, not cooked out of all recognition in some geological event, come to the surface again, not get jackhammered to dust for next door's swimming pool, AND get found by someone who knows what to look for and identifies it as a cambrian annelid....
Does that help?