AJAXSLT is an implementation of XSL-T in JavaScript, intended for use in fat web pages, which are nowadays referred to as AJAX applications. Because XSL-T uses XPath, it is also an implementation of XPath that can be used independently of XSL-T.
A regular expression (regex) is a pattern that can be used to match a string of text. They are a standard feature of many programming languages that are used for text-processing purposes (and they have the inconvenient habit of being implemented ever-so-slightly differently in different languages).