Hi Seork,
Nofollow is an attribute that can be applied to links to ensure inform search engines not to follow the link.
They were initially used to stop people spamming blogs by placing multiple links to their sites in comments sections. The nofollow attribute tells search engines not to place any value on the outbound link from the page.
More recently it has been used to cnotrol the flow of pagerank through a site. Webmasters can then ensure that the pages they wish to rank well in the search engines have the highest pagerank on their site. The formulation of a nofollow lnik can be seen in the example below:
<a href="www.example.com" rel ="nofollow">Example</a>
The nofollow attribute can also be used in the meta robots tag of a page to tell search engines not to follow any of the links on the page. This is used less frequently but can be used on pages where there is a necessity to have a large number of outbound links. An example of meta nofollows is shown below:
<meta name="robots" content="nofollow">
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Broken links are a completely different thing. This is here you have a link on your site that does not function correctly. The most common place for a broken link to end up is on a 404 page but broken links can also cause DNS errors or point to incorrect content on another site.
There are a number of programs that will help you out with detecting broken links. My favourite is Xenu.
Any more questions, you're in the right place!