Bed bugs (sometimes spelled bedbugs, without the space) are fairly small (1/8"-1/4"), wingless, reddish-brown insects that are only found in association with people, usually in the room where people sleep. You never find a bed bug just crawling around outside, for example! They don't move very fast, can't fly and depend on us to move them from place to place usually by hiding in our belongings.
Bed bugs are normally first detected because of mosquito-like bites that occur during the night. Bites itch but are not otherwise dangerous because bed bugs do not spread disease like some other insects. Once bites occur an inspection of the room and bedding will usually turn up the insects between the mattresses, on mattress seams or on bedside furniture.
See pictures of bed bugs and a drawing of hiding places and bites. The second link also has information about safe and effective ways of getting rid of bed bugs if they do turn up.
Subject: What Do Bed Bugs Look Like?