Howdy VitalOne,
Intelligent Design says that somethings are best explained as the result of an intelligent cause (which could or could not be what people term God) instead of the result of natural selection.
Let's call a spade a spade.
"Intelligent Design" is simply another word for "Creationism". Nothing more, nothing less. Both use the false argument "X is too complex to have come about by natural causes (that I know of), therefore it was magically created". The difference between ID and regular creationism is that ID tries to hide the name of the Grand Old Designer (GOD).
More specifically, ID concentrates two debunked ideas, Irreducible Complexity (ala Michael Behe), and Specified Complexity (ala William Dembski). We can go into more detail regarding these two items if you'd like.
Like someone discovering an ancient civilization and looking at the building structures and saying that they were intelligently designed rather than simply natural formations.
You will note that science up to this point hasn't needed a theory of intelligent design to determine that buildings and other artifacts were intelligently designed. This is because the default method of determining what is "natural" and what is "designed" is by comparison to known natural and designed things. This method, while not perfect, works pretty well.
Unfortunately for ID followers, life and biology has long been classified by science as "natural" so there really isn't any controversy regarding "design" in actual scientific circles. The controversy is only found in political, religious, and sociological circles; invariably created by those who just can't accept the findings of science.
If no fossils are ever found linking different species then an intelligent cause would have to be the inevitible conclusion or a very great possibility. Otherwise you would have to conclude that some how 100 million DNA bases just changed in a few mutations....
Really? Darwin formulated his theory of natural selection by studying the forms of living animals, not fossils. He didn't even know about genetics! Today we can determine genetic relatedness of living things by comparison of DNA (much like a paternity test) without a single fossil required. Fossils are simply one line of evidence, genetics and morphology are two others. Unfortunately for the anti-evolution crowd, all three of these lines of evidence point to the same conclusion: the evolution by natural processes of life on Earth.
Regarding the dichotomy you give above, I don't think that lack of understanding of physical processes really leads to the logical conclusion of a mysterious designer. Your mileage may vary.
The actual supporters of ID don't support Intelligent Design being taught in schools because it would become political and hinder a fair and open discussion of if its really true or not, appearing only to be a religious idea...
That is blatantly false. The Discovery Institute is a main player in ID, and they want it taught in schools. If there really is anything to ID, then it will eventually be taught in schools, because it will have proved itself a science. What proponents of ID have given us so far, however, are sciencey-sounding names, bad reasoning, and lots of hand waving. No science so far...
So whats wrong with Intelligent Design in biology? What makes it false?
See the above paragraph. If you are still interested, we can delve into specifics such as Irreducible Complexity.