If you've ever swapped one of your toys with a friend in return for one of their toys, you have bartered. Bartering is trading services or goods with another person when there is no money involved. This type of exchange was relied upon by early civilizations. There are even cultures within modern society who still rely on this type of exchange. Bartering has been around for a very long time, however, it's not necessarily something that an economy or society has relied solely on.
Barter-based economies are one of the earliest, predating monetary systems and even recorded history. People can successfully use barter in many almost any field. Informally, people often participate in barter and other reciprocal systems without really ever thinking about it as such -- for example, providing web design or tech support for a farmer or baker and receiving vegetables or baked goods in return. Strictly Internet-based exchanges are common as well, for example exchanging content creation for research.
It’s hard to answer that without actually seeing a modern gift economy in action. Luckily, modern gift economies actually do exist. On a small scale, they exist among friends, who might lend each other a vacuum or a cup of flour. There’s even an example of a gift economy on a much larger scale, albeit one that’s not always in operation: The Rainbow Gathering, an annual festival in which about 10,000 people gather for a month in the woods (it rotates among various national forests around the country each year) and agree not to bring any money. Groups of attendees set up “kitchens,” in which they prepare and serve food for thousands of people every day, all for free. Classical economists might guess that people would take advantage of such a system, but, sure enough, everyone is fed, and the people who don’t cook play music, set up trails, teach classes, gather firewood, and perform in plays, among other things.