@Kian's answer mentions using a lexer, but in terms of algorithms I think you'll want to use recursion. HTML is after all a recursive structure:
<div>
    <div>
        <div>
        </div>
    </div>
</div>
Here is a naive JS example - although it's not a complete implementation. (I've included no support for <empty /> elements; for <!-- comments -->; for &entities;; for xmlns:namespaces... writing a full fledged HTML or XML parser is a huge undertaking, so don't take it lightly)
This solution notably skips over the process of lexical analysis, but I've deliberately omitted that to contrast my answer with @Kian's.
var markup = "<!DOCTYPE html>\n"+
             "<html>\n"+
             " <head>\n"+
             "   <title>Example Input Markup</title>\n"+
             " </head>\n"+
             " <body>\n"+
             "   <p id=\"msg\">\n"+
             "     Hello World!\n"+
             "   </p>\n"+
             " </body>\n"+
             "</html>";
parseHtmlDocument(markup);
// Function definitions
function parseHtmlDocument(markup) {
    console.log("BEGIN DOCUMENT");
    markup = parseDoctypeDeclaration(markup);
    markup = parseElement(markup);
    console.log("END DOCUMENT");
}
function parseDoctypeDeclaration(markup) {
    var regEx = /^(\<!DOCTYPE .*\>\s*)/i;
    console.log("DOCTYPE DECLARATION");
    var matches = regEx.exec(markup);
    var doctypeDeclaration = matches[1];
    markup = markup.substring(doctypeDeclaration.length);
    return markup;
}
function parseElement(markup) {
    var regEx = /^\<(\w*)/i;
    var matches = regEx.exec(markup);
    var tagName = matches[1];
    console.log("BEGIN ELEMENT: "+tagName);
    markup = markup.substring(matches[0].length);
    markup = parseAttributeList(markup);
    regEx = /^\>/i;
    matches = regEx.exec(markup);
    markup = markup.substring(matches[0].length);
    markup = parseNodeList(markup);
    regEx = new RegExp("^\<\/"+tagName+"\>");
    matches = regEx.exec(markup);
    markup = markup.substring(matches[0].length);
    console.log("END ELEMENT: "+tagName);
    return markup;
}
function parseAttributeList(markup) {
    var regEx = /^\s+(\w+)\=\"([^\"]*)\"/i;
    var matches;
    while(matches = regEx.exec(markup)) {
        var attrName = matches[1];
        var attrValue = matches[2];
        console.log("ATTRIBUTE: "+attrName);
        markup = markup.substring(matches[0].length);
    }
    return markup;
}
function parseNodeList(markup) {
    while(markup) {
        markup = parseTextNode(markup);
        var regEx = /^\<(.)/i;
        var matches = regEx.exec(markup);
        if(matches[1] !== '/') {
            markup = parseElement(markup);
        }
        else {
            return markup;
        }
    }
}
function parseTextNode(markup) {
    var regEx = /([^\<]*)\</i;
    var matches = regEx.exec(markup);
    markup = markup.substring(matches[1].length);
    return markup;
}
Ideally each of these functions would map very closely onto the grammar defined in the XML specification. For example, the specification defines an element like so:
element    ::=    EmptyElemTag | STag content ETag
... so ideally we'd want the parseElement() function to look more like this:
function parseElement(markup) {
    if(nextTokenIsEmptyElemTag) { // this kind of logic is where a lexer will help!
        parseEmptyElemTag(markup);
    }
    else {
        parseSTag(markup);
        parseContent(markup);
        parseETag(markup);
    }
}
... but I've cut some corners in writing my example, so it doesn't reflect the actual grammar as closely as it should.