HTTP status codes are issued by servers in response to client requests. There are 5 classes of status codes: 1xx for informational responses, 2xx for successful requests, 3xx for redirections, 4xx for client errors, and 5xx for server errors. Common codes include 200 for OK, 301 for moved permanently, 403 for forbidden, 404 for not found, and 500 for internal server error. The 403 and 404 codes are explained in more detail, with 403 occurring when permission is lacking and 404 when the requested page cannot be found. Suggested fixes for 403 include checking file permissions while fixes for 404 involve ensuring the requested URL is correct.