DeepWeb is part of World Wide Web that is not readily accessible by everyone because it is not indexed by popular search engines like Google, Bing etc. The term 'ClearNet' is used for general part of WWW which we use commonly everyday and it is only 4% of whole Internet.
DeepWeb/ DarkWeb is not accessed by normal Internet Browsers. You have to use TOR browser for that and all the websites on DeepWeb don't have normal domains like '.com' but most of them have '.onion' domains.
How can you surf this certain of Internet when you can't easily explore or search through it?
There are some limited Deep Web search engines available varying in different domains.
1- Not Evil
Quite an interesting name for search engine because it does not tracks or keep record of your history.
2- DuckDuckGo
This specific search engine lets you do search on both Webs, ClearNet and DeepWeb.
3- Torch
Alongwith search engine, this one is browser too.
4- Ahmia
Another search engine with quite simpler interface.
5- HiddenWikki
This is list of most popular and different type of Onion based Websites.
You have to tread carefully though because all the rumors about DeebWeb are mostly true because after all its is Uncensored part of Internet.