Base excision:
Involves the removal of a particular damaged nucleotide base from the DNA sequence. The damaged base is first marked chemically for removal, then is later excised out from the sequence by an enzyme called DNA glycosylase. The region is acted upon by another enzyme that removes further bases above and below region of interest. Later, the DNA polymerase resynthesizes the nucleotides which are then ligated by the enzyme called ligase.
Nucleotide excision:
The nucleotide excision system positions itself at the region of the genetic mutation or defect and it then cuts a set of nucletoides from the DNA sequence on either side of the mutation. Once this is completed, the DNA polymerase moves in to synthesize the region that is lost and then ligase takes care of incorporating the synthesized region into the DNA strand.
Mismatch Repair:
Mismatch repair mechanisms kick in when the base pairing rule is violated and bases are paired improperly. In this case the decision as to which base is the legitimate one is done by looking for a chemical modification called "methylation" which is done only to the template strand of the DNA from the child strand replicates. Once the correct base has been determined, the whole region of nucleotides from the region at the methylation mark to the region of mismatch are removed. DNA polymerase aids in resynthesizing the strand.
Suggested Reading:
http://www.nih.gov/sigs/dna-rep/whatis.html
Cellular machinaries for chromosomal DNA repair,Genes & Development, 2004.