"The strangest thing about the Theory of Evolution is that so many people think they understand it."
Darwin's book came out in 1859, but we didn't learn about DNA until 1948. So, the science has definitely moved on, and the definitions did too. To make it worse, some dictionaries have terrible definitions. I'll try to do better.
The first word we need is observation. Observations are the raw evidence. They are directly known, and usually not open to debate. A fact can be an observation, but it can also be something that we know indirectly. For example, ocean tides are an observation, and it is a fact that the Moon and Sun cause the tides. Something can earn the title of "fact" when the evidence for it becomes strong enough. Gravity causing the tides has been a fact for centuries. But notice that all scientific knowledge is by definition tentative and provisional, so a fact that is not an observation does have some faint chance of being overturned someday.
A scientific theory is a set of ideas that tries to tie a body of knowledge together. Much like "music theory". The word proof is more squidgy. Only mathematicians have absolute proofs. In science, the word is used in a more relative sense, to mean that some evidence is really pretty strong.
Next, we have to separate biological evolution from other kinds of evolution. When a chemist says that a chemical reaction evolved, he is only saying that something changed as time passed. He really truly wasn't saying anything more than that. Similarly, you might hear that a political situation has evolved, or that stars and galaxies evolve. None of these have anything to do with biology. The Big Bang is irrelevant to biologists. If it were to be disproved tomorrow, they wouldn't care.
So, we need a definition that applies only to living things. And it shouldn't drag in theories, such as Darwin's theory, because that would be circular. And it should define something measurable.
To quote a famous biologist:
Biological evolution ... is change in the properties of populations of organisms that transcend the lifetime of a single individual. The ontogeny of an individual is not considered evolution; individual organisms do not evolve. The changes in populations that are considered evolutionary are those that are inheritable via the genetic material from one generation to the next. Biological evolution may be slight or substantial; it embraces everything from slight changes in the proportion of different alleles within a population (such as those determining blood types) to the successive alterations that led from the earliest protoorganism to snails, bees, giraffes, and dandelions."or, to quote someone less famous, but more succint:
-- Douglas J. Futuyma in Evolutionary Biology, Sinauer Associates 1986
evolution can be precisely defined as any change in the frequency of alleles within a gene pool from one generation to the next.An allele is just a specific variation of a gene. We all have genes for eye color, but we aren't all the same, because some people have one allele, some have another.
-- Helena Curtis and N. Sue Barnes, Biology 5th ed. 1989 Worth Publishers, p.974
That definition isn't perfect, and some people complain about it, but we'll use it for the moment.
So, the observation that a gene pool has changed, means that one has observed biological evolution. And then we would want a theory to explain or predict that change.
Darwin's theory was about The Origin of Species. So, what is a species? That's actually a deep subject, but the short answer is: a species is a set of creatures which, in their natural setting, breed among themselves.
If a new species comes into existence, we would call that speciation - the opposite of extinction. If species A gives rise to species B and C, then A is the common ancestor of B and C. (This in no way implies that A is extinct. B and C are quite likely to have arisen from geographically isolated groups of A-s.)
Darwin said that evolution and speciation happened because of natural selection, sometimes incorrectly called survival of the fittest. The point is not at all who survives: the point is that the variations between individuals may change how many descendants they get to have. Survival of the fittest has incorrectly been accused of being a tautology. Regardless, phrased as differential reproductive success, it is not circular at all.
Darwin said that natural selection worked because creatures have variation. It being 1859, he didn't know why there was variation. Today, we know that it comes from the many possible combinations of genes. And, it comes from mutations, which are caused by mis-operation of the mechanisms that copy genetic information.
Finally, Darwin said that natural selection had gradually caused all lifeforms on Earth to have evolved from a single common ancestor. That is more or less Darwin's Theory of Evolution. Now, 140 years later, we have a Modern Synthesis of the Theory of Evolution. It's fairly similar to Darwin's, but richer and more detailed. And the emphasis has shifted away from organisms and individuals and gradualism. In the 1930's it became more mathematical, and the ideas became more about genes and populations. In the last few decades, there has been new emphasis on symbiosis and on mass extinctions.