Simply put; the theory proposes the extant (and extinct) species of the world all evolved over time, and still is, by natural selection. The explanation for such changes are that when one individual gain a trait that increases it's chances of survival, it will more likely mate and have offspring who carry the favorable trait. The offspring will also reproduce more successfully, and so the trait will become dominant in a population after some amount of time.
The theory is supported by modern science, both in the fossil record and even by observation.
Sadly, Darwin could not explain all that he observed. A monk called Gregor Mendel had more or less discovered genes at about the same time, explaining how evolution works.
The theory is extensive, for more information surf the grat internet (just avoid the creationalists who usually has not read much about it) or even read one of Darwins books; especially "on the origins of species".
The theory of evolution observed
Q: Why do antibiotics stop working after a period of use?
A: Because bacteria are very good at adapting. Once one of the bacteria obtain resistance against the antibiotic, it will multiply (replacing the ones killed by the antibiotic) and even give the other bacteria the "recipe" (DNA)
Another one for you: a group of scienists measured the beak size of the birds on a secluded island over several years. Then a storm came, not harming the birds, but reducing the number of small seeds, leaving big seeds the major food-source for the birds. When they measured beak size of the offspring, the average size had increased - indicating the ones with larger beaks better at crushing large seeds survived and reproduced.
And another one: Modern genetics can analyze DNA (thats how you can detect who are related to each other in a family, and also find a killer from a strand of hair he left behind). When comparing DNA it is easy to see the DNA of modern humans and modern orangutans and other apes are extremely similar.