When a note gets redeemed, it usually gets cancelled so that it can't be spent anymore. These cancellations on the notes were done differently depending on the institution from which the note came. The Confederate States for example, would punch large holes in their notes as shown below.
$10 Confederate States Of America (1862)
Notes issued by the Government of Texas on the other hand, would cut cancel the notes. This allowed the notes to be cancelled by one can still appreciate the beauty of the note as shown in an earlier post on Texas. The Texas Treasury warrants however, would have X's drawn on them.
Some of the more interesting cancellations can be found in the Bible Belt, namely North Carolina. The Bank of Washington, NC would write the word "Judgement" on the note since the note is no longer in circulation. It is as though the note has died and had judgement passed on it.
The Moravians on the other hand, were a highly religious community which settled in Salem, NC (Now part of Winston-Salem) back in the 1700's. The Salem branch of the Bank of Cape Fear, rather than passing Judgement on a note when one returned it to the bank, would have a red cross put on the note to indicate that it has been redeemed.
$3 Bank Of Cape Fear, branch at Salem, NC (1860)