Damerau-Levenshtein distance is an extension to Levenshtein distance. It is also defined as minimum number of simple edit operations on string to change it into another, but the list of allowed operations is extended.
As it is written on Wikipedia there are 4 allowed edits: deletion, insertion and substitution of an single character and an transposition of two adjacent characters.
Example. Such defined distance between words gifts and profit is 5:
gifts => pgifts (insertion of 'p') pgifts => prgifts (insertion of 'r') prgifts => proifts (substitution of 'g' to 'o') proifts => profits (transposition of 'if' to 'fi') profits => profit (deletion of 's')