Internet Explorer is the main reason why doctypes are important
The DTD (Document Type Declaration) is there to tell a browser how the page is to be rendered (displayed), what tags are valid in the source code, what attributes exist for the tags, the level of scripting and the valid events and methods that exist for that page model.
It is the code template that HTML pages using that doctype should conform to.
There is a list of current and valid DTDs at ->
http://www.w3.org/QA/2002/04/valid-dtd-list.html
Without a DTD or a with an incorrectly defined one, browsers render in what is known as "quirks" mode.
This means that a default method of rendering will be applied and each browsers default are different, and Internet Explorer's quirks mode is
very different indeed. In this mode everything it can do wrong, it does do wrong.
More on this from ->
http://www.positioniseverything.net/ie-primer.html
Adding a valid doctype makes IE run in standards mode, where it's
deliberately defined bad behaviour is curtailed somewhat, and only the very many bugs in IE6 remain to vex the unwary, IE7 fixed many of the "bugs" but besides leaving some also introduced a few new ones
An IE buglist ->
http://www.positioniseverything.net/explorer.html
An Opera one ->
http://www.positioniseverything.net/op-omnibus.html
and a Gecko browser buglist ->
http://www.positioniseverything.net/gecko.html
So the bottom line is:-
If you are not bothered about how your pages or designs will appear to people using different browsers, don't worry about using a DTD. But on the other hand, if you would want everyone who views the page to see the same thing regardless of their choice of viewing agent (albeit with a few relatively minor differences) the use an appropriate DTD for your code level.
As designers, developers and code monkeys, "we" will always advocate the use of a doctype, because we know from experience the majority of cross-browser "problems" with a layout will disappear if one is used from the outset.