The answer is an:
HTTP 404 Error Page
I appear if things are broken,
If attempting to access a site, and the page is not found, or the server is not running something (or various other reasons), a 404 error page will appear.
I'm not there if things are flawless,
While not necessarily true, if you don't get a 404, I think the OP is referring to the chance that you're getting a 200, indicating that everything is working correctly.
But you can track me,
It's possible to use tools that crawl webpages to track dead links, or, in the case of a live site, like a forums, will check the link as the page is rendered. (Often via a cache)
I'm not wanted,
No one wants to get the dreaded 404 error page.
Apparently 404 has become slang for
clueless in the UK.
I'm different to all,
Most every site has a different 404 error page (unless using something like an initial PHP or IIS install, and they have not changed it.)
You can't touch me by hand.
You...can't touch a 404 error page. Unless you print it out and no one wants to know about that, anyways.
I can go from 0 to 20.
The HTTP 404 error has 20 sub-statuses ranging from 404.1 to 404.20.
Some websites report a "not found" error by returning a standard web page with a "200 OK" response code, falsely reporting that the page loaded properly; this is known as a soft 404.
Humor is used on me (but not always)
Many 404 pages are humorous to lighten the mood of getting a dead link.