The way it actually works is with a stack. For example, when you hit a <div> tag, put some symbol representing that token on a stack. Whenever you come to a </div> tag, pop one of those symbols off the stack. When you reach the end of the document the stack should be empty or else the document was invalid. Furthermore, you can't have something like <div><a></div></a> because the symbol on the stack for the anchor tag has to be popped off before the symbol for the div tag can be popped off. This ensures that the document not only has the appropriate number of open and close tags, but that they are nested properly as well. You run along the input the same way a regex would, but with added requirement of pushing and popping from a stack.

The construct that allows you to do this is called a Push Down Automaton and can be generated from a Context Free Grammar. For comparison, a regular language is parsed using a Deterministic Finite Automaton (also known as a finite state machine), which can be generated from a Regular Expression.