Bed bug bites look a lot like bad mosquito bites. Take a look at the lower drawing (left). Bed bug bites will generally be red with local inflammation and swelling. The redness is caused by an allergic reaction to proteins in the bug's saliva; again, much like a mosquito bite. There may be multiple bites in the same area and the bites can occur anywhere on your body. Bites will usually itch as well.
Bed bug bites are self-limiting and there is very little danger of disease transmission associated with these insects (unlike mosquitoes). Bites can be treated with over-the-counter anti-itch medications or antibiotic lotions if the skin is broken by scratching.
If you encounter bed bugs in a motel/hostel/inn or other sleeping establishment you should (1) notify the management so they can take steps to control the infestation, and (2) take care not to transport bed bugs from there to your next destination. Bed bugs can travel with you on your belongings such as backpacks, bedding, sleeping bags, pillows, etc.
Notice in the upper drawing that bed bugs like to hang out in the folds and seams of the mattress and under the mattress pad (red arrows). The black specks in this drawing are bed bug droppings, mostly digested blood. Rooms that are infested with bed bugs will usually harbor this type of evidence of the bug's presence.
See the following links for information about the life cycle of bed bugs and eliminating bed bug infestations. There's also some information about checking a room for bed bugs before moving in for the night.