For argument's sake lets assume a HTML parser.
I've read that it tokenizes everything first, and then parses it.
What does tokenize mean?
Does the parser read every character each, building up a multi dimensional array to store the structure?
For example, does it read a
First of all, you should be aware that parsing HTML is particularly ugly -- HTML was in wide (and divergent) use before being standardized. This leads to all manner of ugliness, such as the standard specifying that some constructs aren't allowed, but then specifying required behavior for those constructs anyway.
Getting to your direct question: tokenization is roughly equivalent to taking English, and breaking it up into words. In English, most words are consecutive streams of letters, possibly including an apostrophe, hyphen, etc. Mostly words are surrounded by spaces, but a period, question mark, exclamation point, etc., can also signal the end of a word. Likewise for HTML (or whatever) you specify some rules about what can make up a token (word) in this language. The piece of code that breaks the input up into tokens is normally known as the lexer.
At least in a normal case, you do not break all the input up into tokens before you start parsing. Rather, the parser calls the lexer to get the next token when it needs one. When it's called, the lexer looks at enough of the input to find one token, delivers that to the parser, and no more of the input is tokenized until the next time the parser needs more input.
In a general way, you're right about how a parser works, but (at least in a typical parser) it uses a stack during the act of parsing a statement, but what it builds to represent a statement is normally a tree (and Abstract Syntax Tree, aka AST), not a multidimensional array.
Based on the complexity of parsing HTML, I'd reserve looking at a parser for it until you've read through a few others first. If you do some looking around, you should be able to find a fair number of parsers/lexers for things like mathematical expressions that are probably more suitable as an introduction (smaller, simpler, easier to understand, etc.)