Mr. Gutwrench is a made-up teacher that Leslie imagined her teacher Mr. Garrison to be like.
Mr. Gutwrench is described by Leslie as "the meanest teacher alive". When the principal puts Leslie in his class in Back to School, Mr. Gutwrench immediately made fun of her torn dress, gave her a 10,000-word essay to write, and yelled at her for hanging out with her friends during the summer. Then, after Leslie made what she called an "innocent little statement", he gave out homework. In It Sure Seems Like It to Me, his antics continue. However, in the same episode, it's established that his actual name is Mr. Garrison, and that his personality is the opposite of how Leslie was describing him — thus proving she was blowing him out of proportion.
Any family he may have is unknown.