What are some algorithms for comparing how similar two strings are?

You could use ngrams for that. For example, transform the two strings in word trigrams (usually lowercase) and compare the percentage of them that are equal to one another.

Your challenge is to define a minimum percentage for similarity.

http://en.wikipedia.org/wiki/N-gram

Damerau Levenshtein distance is another algorithm for comparing two strings and it is similar to the Levenshtein distance algorithm. The difference between the two is that it can also check transpositions between characters and hence may give a better result for error correction.

For example: The Levenshtein distance between night and nigth is 2 but Damerau Levenshtein distance between night and nigth will be 1 because it is just a swap of a pair of characters.

What you're looking for are called String Metric algorithms. There a significant number of them, many with similar characteristics. Among the more popular:

• Levenshtein Distance : The minimum number of single-character edits required to change one word into the other. Strings do not have to be the same length
• Hamming Distance : The number of characters that are different in two equal length strings.
• Smith–Waterman : A family of algorithms for computing variable sub-sequence similarities.
• Sørensen–Dice Coefficient : A similarity algorithm that computes difference coefficients of adjacent character pairs.

Have a look at these as well as others on the wiki page on the topic.