What exactly is evolution? Evolution is a theory. It is not a hypothesis.  A theory is a set of scientific principles that is supported by observable facts. A hypothesis, by contrast, is a statement not backed up by facts. 

The theory of evolution is as follows:

Fact #1: Random mutations occur in nature.

Fact #2: Mutations tend to be passed onto off-spring.

Fact #3: If one organism is genetically different enough from another organism then they will not be able to reproduce together.

Therefore:

Traits that benefit an organism (or at least don't kill it) will be passed onto its offspring.  Over millions of years, the accumulation of these changes, both big and small. will result in different species based on the local geographic conditions.

For example, a bear that spends most of its life swimming around in the water gives birth to offspring with clubbed feet. This bear is able to swim better as a result than other bears and is thus more likely to catch fish and therefore more likely to survive to pass on its genes. Over time, all the bears in that geographic area will have clubbed feet.

By contrast, same bear, 10 miles inland, born with same clubbed feet is no longer able to hunt food in the woods. It dies before it can reproduce. And with its death so too does that genetic trait.

Our water-loving, club-foot bears, by contrast, have now specialized (or adapted) to life in the water.  Even though they are, at this point, still the same species as their wood-prowling cousins, they no longer interact/interbreed because they spend all their time in the water.  But they are still the same species at this point.

Time passes. More genetic mutations occur over a very long period of time. Other changes are simply the result of certain genes that are already part of the bear being more survivable.  Living in the water all the time has fewer size restrictions. Larger bears are thus not at a hunting disadvantage as their forest living cousins are. So they get bigger and bigger as subsequent generations pass.  Bears, which are fatty to begin with, get fattier still because there's no disadvantage to it and those that are fatty can survive in the cold water better.  One day a genetic mutation results in offspring that has no hair.  The water-swimming club-footed, fatty large bear would have died if the mutation had occurred 50,000 years before when they didn't have all the fat on them from the cold water.  But now it's no disadvantage at all.  In fact, this bear can swim even faster thanks to not having hair dragging it down.  Those genes pass onto their off-spring.

Repeat the process over 50 million years and the genetic differences create a new species.  It's still basically a bear but it can no longer breed with its forest-living ancestors.  It doesn't look like a bear though. It's hairless and its arms are clubbed and its tail looks like a big old fin now and its hind legs are just vestigial.  One day, a human spots one in the water and gives it a name -- whale.

Of course, there are also various species that are in-between those stages or have gone off in their own directions.  After all, different environments favor different traits.  We have walrus's, seals, etc.  They took different paths from one another and became their own species.

What evolution is NOT:

Evolution does not talk about how life came to be on Earth in the first place.  There are no theories on how life started on Earth.  There are numerous hypothesis's on it. Maybe amino acids got together in some random way and pretty soon you have the earliest forms of primitive life.  Perhaps the first single celled life forms were seeded from aliens (which only begs the question, where did they come from?).  Maybe God or some other super-natural force created the first spark of life.  We don't know. We may never know (though my bet is that they'll find a way to create life in a lab at some point from inorganic molecules).

Evolution vs. Intelligent Design

Intelligent Design is a new hypothesis (or at least a new name for creationism). It argues that God or an alien force created life on earth and then has directed its "evolution" on earth.   It is not science.  It's not science because nothing in it can be scientifically tested or proven.  By contrast, evolution can and has been tested.  Even before we had DNA tracking technology, evolution had been well documented in countless cases (particularly in the case of Pacific Island birds where we can see a single species of bird turning into hundreds of species of different birds of different genus's in the course of hundreds of thousands of years).  Today, with the ability to look at Mitochondrial DNA we can literally graph where different species evolved from and how long ago it occurred.

What about "Missing Links"?

Discussion of missing links and the like do nothing to disprove evolution, as a theory.  When someone says "There is a hole in the fossil record" one might say "Well of course there is."  Missing links were more interesting before mitochondrial DNA research became widespread.  Barring a time machine, no one can say for 100% that evolution is how we have all the different types of animals we have today.  But it is, by far, the strongest explaination we have.