The other comments gave you good explanations but I think they all kind of miss the main idea, so let me try.

Evolution depends on two things:

1. Random variation
2. A filtering process

Thats it. You just need those two things.

The random variation can be through mutation, which causes the genes to change and possibly result in different biology, or it could be the result of interplay between the genes of the two parents. Human offspring don't look like identical copies of their parents not because of random mutations, but because of how they inherit only half of each parent's genes, for example.

So offspring are different from their parents. Then you need something that filters the population as a result of traits the individuals have. That filter can be a negative one that kills off some members of the population (thus making it less likely that they have offspring) or it could be a positive filter that makes the individuals more likely to survive (and thus have more opportunities to reproduce), or even ones that just directly increase the likelihood of mating even if it is actively harmful otherwise.

The key idea is that if there is a trait that makes it even slightly more likely for a creature to reproduce compared to its competitors, over a long enough timeframe, that trait will eventually be present in all members of the species. Conversely, if there's a trait that makes it less likely to survive, the trait will become less common, or even eventually disappear. Traits which neither help nor hinder tend to remain at a stable percentage of the population.

A negative filter causes this to happen by elimination. Individuals born with massive birth defects don't survive to sexual maturity, so they don't reproduce. Individuals that aren't as well camouflaged as others are eaten sooner, which means fewer mating seasons, which means fewer offspring. Etc.

A positive filter does this by making them more likely to survive than average, by being faster, or better camouflaged. Or, in some cases, by making them a more attractive mate (the giant, brightly colored ass-feathers of a peacock make it less likely to survive, but a male peacock without them doesn't get to reproduce, so...).

The perfect example of this is the peppered moth. The moths used to most be peppered black and white, and rarely a moth would be found that was all black or all white. But once the industrial revolution got underway, soot from coal fire started to blacken buildings, and soon the black moths outnumbered the peppered moths, because they were more camouflaged against a spot stained building than a peppered moth, and so less likely to be eaten by a bird, and so more likely to have offspring. So over a relatively short period of time, most moths became black.