Evolution is the change in allele frequency in a population of organisms over many generations. An allele is a version of a gene. Blue eyes, green eyes, and brown eyes are different alleles of the eye color gene. For example, we might see a particular shade of fur become more common in a species population that has migrated to a new environment with a new palette of natural background colors.
The theory of evolution is a conceptual model that considers the implications of observable evolution occurring over vast amounts of time, which lets us accurately predict future experimental data like where a certain transitional fossil will be found that we haven't discovered yet, or how long it will take for a disease to develop resistance to an antibiotic. It has made hundreds of thousands if not millions of accurate future predictions, and is one of the most reliable theories in science.
It is very easy to observe evolution. You can take a bit of soil from your backyard with billions of bacteria in it, and place it in a container of nutrient broth that has very low levels of one vital ingredient (like poor amounts of any chemical containing carbon for the bacteria to use). Then, add a substitute for that ingredient that no bacterium can digest, like a new pesticide. Slowly increase the concentration of this chemical over time while growing the bacteria, and you'll see the evolution of a new metabolic pathway to digest this new chemical. You can sequence the DNA of the bacteria at the start and the end of this experiment, and see the genetic changes that result in a new set of enzymes to break up the chemical and transform it into something usable. This has been done many many times, and provides a strategy to clean up chemical spills, and is evolution.
When England was experiencing its industrial revolution, we saw moths evolve a darker color because the trees they perched on were becoming coated with soot. Any fruit or vegetable you can buy at the store is a product of evolution guided by humans. There are endless verifiable and repeatable examples.
Some people follow religions whose interpretations or doctrines require the earth to be extremely young. Otherwise, the internal logic of their beliefs breaks down. Without an Adam and Eve, for example, there would be no original sin, and there would be no need for Christ to redeem the sins we are born with and no need for his supposed death and resurrection. The theory of evolution directly refutes the possibility of an Adam and Eve. And so we see this cottage industry of apologetics to deny, miseducate, obfuscate, and selectively cherry pick information about evolution. One example is the "microevolution" and "macroevolution" distinction, which is meaningless to biologists because it's all the same evolution.
