Charles Darwin, known as one of the fathers of modern evolutionary theory, defined evolution as an ongoing process of descent with modification. He theorized that certain factors and pressures influence which organisms will survive and reproduce, thus passing on whatever traits allowed them to survive in those conditions.
It's this process that encompasses evolution. The theory of evolution is what causes organisms to diversify to fit into various ecological niches and to develop characteristics that allow them to survive and reproduce. Evolution is the gradual and cumulative changes that an organism undergoes throughout time.
Darwin also posited that there are certain processes that allow evolution to occur. Without these processes, evolution would, essentially, not exist as we know it.
Process One: Natural Selection
Natural selection is perhaps the main driving force of evolution. In fact, most people refer to evolutionary change as "evolution by way of natural selection."
In order to understand natural selection, three things must be understood.
First is that every population of organisms is going to have variation in their traits. For example, a population of field mice may appear tan, brown, and white.
Second is that many of these traits are heritable. This means that parents will pass whatever traits they have down to their offspring when (and if) they reproduce.
The third thing to understand is that reproduction isn't guaranteed or equal for every member of the population. Back to the field mouse example, not all mice are going to be able to find mates, survive past their early months, be healthy enough to reproduce, etc.
Now that those facts are clear. In short, natural selection is how certain traits, characteristics, and behaviors within organisms are "selected" by the environment as advantageous. When an organism has a trait that is advantageous, it will help that organism survive in the environment. This allows them to survive and reproduce, thus passing down that advantageous trait to the next generation.
Organisms without that trait are less likely to survive and reproduce, meaning there will be more organisms in the next generation with that trait than without (since organisms without won't be able to reproduce and pass on their trait). Thus, the advantageous traits are naturally "selected" to become standard in the population, which leads to the evolution of the species as a whole over time.
Take the field mice, for example. Let's say you have a population of mice with varied colors of tan, brown, and white.
White field mice are going to be easily spotted and preyed upon by predators. Thus, the "white" trait will not be passed down to the next generation. Tan and brown mice, though, will easily be camouflaged, which will help them avoid predation. This means that they'll pass their genes for that trait onto the next generation, which drives the evolution of the mice to be (primarily) tan/brown.
This is a simple example, but it gives the general idea of the process.
Process Two: Artificial Selection
Artificial selection is the same general process as natural selection with the difference being that humans artificially select which traits they desire to be fixed into the population instead of the traits being selected by nature/the environment. It's also referred to as selective breeding.
Artificial selection is the intentional selection of the parental organisms in order to create offspring that have the beneficial or desired traits that the parents have.
For example, many farmers will "select" the strongest horses to reproduce in order to get horses that are overall strongest. Or they'll select cows that produce the most milk to reproduce in order to get offspring that also produce more milk.
This can also be done with plants. For example, one may choose parental organisms that produce the most fruit or the largest flowers.
Process Three: Microevolution
Microevolution is defined as small scale evolutionary processes where the gene pool of a certain species (or a single population of a species) is changed over a short period of time. Microevolution is usually the result of a natural selection, artificial selection, genetic drift and/or gene flow.
Process Four: Macroevolution
Macroevolution happens over extremely long periods of time, unlike microevolution. Also unlike microevolution, it happens on a much larger scale. Instead of a single population, it could affect an entire species or subset of species in a specific order.
Common examples of macroevolution include the diverging of one species into two distinct species and the culmination/combination of many instances of microevolution over time.
Process Five: Coevolution
Coevolution occurs when the evolution and natural selection of one species has a direct effect on another and leads to the evolution of that other species.
For example, let's say a bird evolves to eat a specific type of bug. That bug may then evolve a defense against that bird like a hard outer shell. This could then trigger the bird's evolution of a beak that allows them to crush the hard outer shell of the bug.
These coevolutions are caused by specific selection pressures that arise because of the evolution of one species. It's often referred to as a sort of "domino effect", which can be seen in the bird-bug example quite clearly.