The basics

A microsatellite consists of a specific sequence of DNA bases or nucleotides which contains mono, di, tri, or tetra tandem repeats.  For example,

AAAAAAAAAAA would be referred to as (A)11
GTGTGTGTGTGT would be referred to as (GT)6
CTGCTGCTGCTG  would be referred to as (CTG)4
ACTCACTCACTCACTC would be referred to as (ACTC)4

In the literature they can also be called simple sequence repeats (SSR), short tandem repeats (STR), or variable number tandem repeats (VNTR).  Alleles at a specific location (locus) can differ in the number of repeats. Microsastellites are inherited in a Mendelian fashion.
Microsatellites are highly variable.  At one locus, you may have 10 repeats, and your friend may have 15 or 20 repeats. Why?
To "evolve" simply means to change.  Microsatellite alleles change (mutate) over time. In a population, there may exist many alleles of a single microsatellite locus.  Microsatellite alleles differ in the number of repeats.  For example, one allele may have 7 repeats of a CT motif, and another allele may have 8 repeats.   In a population, there may exist many alleles (up to 70 or 80!) at a single locus, with each allele having a different length. An individual who is homozygous for a locus will have the same number of repeats on both chromosomes, whereas a heterozygous individual will have different numbers of repeats on the two chromosomes. The regions surrounding the microsatellite locus, called the flanking regions, may still have the same sequence.  This is important because the flanking regions can therefore be used as PCR primers when amplifying microsatellite loci, and can be conserved across genera or sometimes even families.  Below, the two lines represent the sequences on two homologous chromosomes in a diploid organism.  (For clarity, only one strand of each chromosome is shown.)

Homozygous: (Both strands have 7 CT repeats)

…CGTAGCCTTGCATCCTTCTCTCTCTCTCTCTATCGGTACTACGTGG…
…CGTAGCCTTGCATCCTTCTCTCTCTCTCTCTATCGGTACTACGTGG…
Heterozygous: (One strand has 7 repeats, and the other has 8 repeats)

…CGTAGCCTTGCATCCTTCTCTCTCTCTCTCTATCGGTACTACGTGG…
…CGTAGCCTTGCATCCTTCTCTCTCTCTCTCTCTATCGGTACTACGTGG…

How were these different alleles created?  Mutation!  Interestingly, it is estimated that microsatellites mutate 100 to 10,000 as fast as base pair substitutions. This makes microsatellites useful for studying evolution over short time spans (hundreds or thousands of years), whereas base pair substitutions are more useful for studying evolution over long time spans (millions of years).

Evolution of Microsatellites:

There are two hypotheses that explain how microsatellites mutate.
1.“Polymerase slippage” or “slipped-strand mispairing.”
 
When the DNA replicates, the polymerase loses track of its place, and either leaves out repeat units or adds too many repeat units.    The result is that the new strand has a different number of repeats as the parent strand.  For an excellent illustration of how this “slippage” may occur, see the article DNA Microsatellites: Agents of Evolution? (Scientific American, January 1999, 94-99).  This is thought to explain small changes in numbers of repeats (adding or subtracting one or just a few repeats).  It also explains how microsatellite loci could be generated in the first place; it is likely that sequences including two or three repeats are randomly distributed throughout the genome.  Slippage could them amplify these short repeat sequences into many repeats over successive generations.  Certainly, the effectiveness of the mismatch repair system would also play an important role in microsatellite mutation rate.  
2. Unequal crossing-over during meiosis.  
This is thought to explain more drastic changes in numbers of repeats.   In the diagram below, chromosome A obtained too many repeats during crossing-over, and chromosome B obtained too few repeats.
Models of Microsatellite Mutation
1. Stepwise Mutation Model (SMM)
This model holds that when microsatellites mutate, they only gain or lose one repeat.  This implies that two alleles that differ by one repeat are more closely related (have a more recent common ancestor) than alleles that differ by many repeats.  In other words, size matters when doing statistical tests of population substructuring. The genetic distance statistic that uses this model is called Rst.  The SMM is generally the preferred model when calculating relatedness between individuals and population substructuring, although there is the problem of homoplasy.
2. Infinite Alleles Model (IAM)
Each mutation can create any new allele randomly.  A 15-repeat allele could be just as closely related to a 10-repeat allele as a 11-repeat allele. All that matters is that they are different alleles. In other words, size isn't important.  The statistic that uses this model is called Fst.
Because microsatellites mutate quickly, they can be used to study recent population evolution, relatedness (by doing a microsatellite "fingerprint" for different individuals), and forensics.