The auto-mode-interpreter-regexp is a variable used by SetAutoMode that determines the MajorMode based on the first line of the file.
For example, if this is the first line of the file:
#!/bin/bash
Then the file is as determined to have an associated interpreter, in this case it is the Bourne again shell, Bash, located at /bin/bash.
Emacs then uses the variable ‘interpreter-mode-alist’ to use the assocations of interpreters (”python”, “awk”, “guile”, “sh”) to modes (‘python-mode’, ‘awk-mode’, ‘scheme-mode’, ‘sh-mode’).