In a home exchange, two households swap homes. The company I usually use is Intervac.com. In a hospitality exchange someone hosts you in their home, and in return, you host them in yours later on. (www.hospitalityclub.com)
In a cultural exchange you stay in someone’s home and teach them about your culture or your language. (usservas.org) These kinds of exchanges are usually free of charge. For small fees ($25 to $100) you can join networks that help you locate prospective swappers.
Other networks like www.couchsurfing.com are free of charge. For a list of over 20 home/hospitality groups, visit the home exchange section of the “favorite links” area of this website.