So how does a HTML parser work? Doesn’t it use regular expressions to parse?
Well, no.
If you reach back in your brain to a theory of computation course, if you took one, or a compilers course, or something similar, you may recall that there are different kinds of languages and computational models. I’m not qualified to go into all the details, but I can review a few of the major points with you.
The simplest type of language & computation (for these purposes) is a regular language. These can be generated with regular expressions, and recognized with finite automata. Basically, that means that “parsing” strings in these languages use state, but not auxiliary memory. HTML is certainly not a regular language. If you think about it, the list of tags can be nested arbitrarily deeply. For example, tables can contain tables, and each table can contain lots of nested tags. With regular expressions, you may be able to pick out a pair of tags, but certainly not anything arbitrarily nested.
A classic simple language that is not regular is correctly matched parentheses. Try as you might, you will never be able to build a regular expression (or finite automaton) that will always work. You need memory to keep track of the nesting depth.
A state machine with a stack for memory is the next strength of computational model. This is called a push-down automaton, and it recognizes languages generated by context-free grammars. Here, we can recognize correctly matched parentheses–indeed, a stack is the perfect memory model for it.
Well, is this good enough for HTML? Sadly, no. Maybe for super-duper carefully validated XML, actually, in which all the tags always line up perfectly. In real-world HTML, you can easily find snippets like <b><i>wow!</b></i>. This obviously doesn’t nest, so in order to parse it correctly, a stack is just not powerful enough.
The next level of computation is languages generated by general grammars, and recognized by Turing machines. This is generally accepted to be effectively the strongest computational model there is–a state machine, with auxiliary memory, whose memory can be modified anywhere. This is what programming languages can do. This is the level of complexity where HTML lives.
To summarize everything here in one sentence: to parse general HTML, you need a real programming language, not a regular expression.
HTML is parsed the same way other languages are parsed: lexing and parsing. The lexing step breaks down the stream of individual characters into meaningful tokens. The parsing step assembles the tokens, using states and memory, into a logically coherent document that can be acted on.