What is incomplete dominance? What is simple dominance?

Remember that the genes encode proteins. If a gene behaves according the the classic dominant-recessive rules, it means one copy of the gene can produce enough protein to do the job of that gene well enough that you can't distinguish it from two copies.

In incomplete dominance, the amount of protein made by transcribing and translating one copy of the gene is NOT enough to give the "full" phenotype. Twice as much protein (made by using two good copies of the gene) provides extra activity that produces a different phenotype. People classically thought this was a rare occurrence, but many human genes are thought to be good enough in one copy, but better in two copies. For example, many genes that code for DNA repair enzymes are thought to be sufficient in one copy, but may increase the rate of cancer causing mutations, even before the second copy is lost through mutation.

Simple dominance is the way that Mendel described it. One form of the gene is dominant over the other form which is called recessive. Using T for tall and t for short:
TT and Tt are both tall. tt is short.

Multiple alleles means that there are three or more forms of a gene for a trait. That doesn't mean that a person can have all three of them. The person still only gets two ... it's just that there are three or more of the alleles in the gene pool that are available.