There's no such thing as a 'dofollow site' really - the only thing that can be 'dofollow' or 'nofollow' is a link. If a link has a rel='nofollow' attribute in its code then search engines will not allocate any Page Rank from that page to the target of the link - making it less useful for SEO purposes. A 'dofollow' link is a link that doesn't isn't marked as nofollow. However if someone is talking to you about a 'dofollow site' they probably mean a site where you can get dofollow links - through blog comments, link buying or similar.
If your are using Firefox download a plugin called "SEO for Firefox". Once installed you can choose to 'Highlight' dofollow links. This way every time you visit ANY website the links will show a colored highlight if it's dofollow.
you can make an entire page or site a nofollow by using meta robot attribute or robot.txt. But usually it is rel nofollow attribute we are taking about. And to check it you can use the firefox plugin already talked about or you can visit http://tools.seo-free.org/link_extractor.php also
You can check do follow & no follow site by any tool. The best tool is mozbar..type in Google MOZBAR & Download it & you will see a button of Nofollows in this bar. Just click on this button & all nofollow links will display in pink colour..
Another good way to check for links with nofollow attribute is to check it manually. View its source code, press CTRL + F, then type nofollow.
Only dofollow links helps in driving more traffic as well as rank your site well in SERPs. On other hand, no dofollow links helps in driving traffic only. You can use NoDoFollow Addons of Firefox to check the links that are dofollow or not.