Depending in large part on where you're from, a hug between friends can be the most natural thing in the world - or an invasion of your personal space. I hug family and good friends, and I think that is typical of the majority of Americans. Of course there are exceptions and variations; some Americans hug and kiss, or just kiss, and I have a few friends who hate hugging, but generally speaking, it's part of our culture. In addition to family and friends, we sometimes hug acquaintances and even strangers, such as to say thank you for an act of kindness or to offer comfort. So it's only natural that when traveling in France, Americans sometimes want to hug people they meet. Natural, but unfortunate.
The French don't hug, at least not as much or for the same reasons as Americans do. Presumably French lovers hug, and parents hug their young children, but I have never personally seen two French people hug, not even married couples (though I saw a movie in which a mother and daughter exchanged a sort of half hug by grasping one another's shoulders prior to a long separation). Suffice it to say that the French very rarely hug, and when they do, it's certainly not a big bear hug or a full body press. And it's most definitely not between strangers, or acquaintances, or even most friends and family. Whether to say hello, good-bye, or thank you, Americans usually hug, whereas the French kiss cheeks and shake hands.
This cultural difference can lead to unpleasant situations. When my mother-in-law, who speaks no French, visited us in France, we spent three days at a bed and breakfast in Arles. The owner was a lovely woman of about 40 who spent a fair amount of time chatting with us, about half in English and half in French. My mother-in-law is an exceptionally friendly, outgoing person, and when we left, she said thank you and good-bye to the owner the only way she knew how: with a big hug. The owner had been expecting les bises, and when my mother-in-law wrapped her arms around her, the owner's arms stayed woodenly at her side. She didn't say anything, but her confusion and even dismay were palpable, and when I then approached to do les bises she leaned back slightly for fear of another onslaught.
But I think this is even more telling: I have a dear friend in Rouen whom I met and chatted with online for about a year before I met her in person, when she invited us to stay with her for a few days. My friend teaches English and is a huge fan of la culture anglo-saxonne, so when she asked me to explain what a "hug" is, I didn't hesitate to give her one. While I wouldn't say she was as shocked as the B+B owner, my friend stammered and blushed, though at least we were able to laugh about it. But the point is that despite her familiarity with the English language as well as English and American culture, she had no idea what a hug was outside of a romantic or familial context, and my demonstration made her uncomfortable.
Bottom line, I strongly urge you to avoid hugging any French people, unless they initiate it - and I can pretty much guarantee they won't. But to answer the question, the closest translations of "to hug" are embrasser (to embrace, but more commonly to kiss), étreindre (to embrace, but also to grasp, seize), and serrer dans ses bras (to hold tightly in one's arms). As for the noun, you can try une étreinte (which can also mean grip or stranglehold) or the literary term une embrassade (which Le Petit Robert defines as action de deux personnes qui s'embrassent amicalement).
Several people have suggested un câlin, but that means "cuddle" rather than "hug" and is definitely something limited to lovers and parents/children. I would never give un câlin to say hello to a friend, or to thank a stranger.
Do you have a story about hugging in France you'd like to share? Whether you're American, French, or another nationality and have given or received an unwelcome hug, we'd love to hear about it: share your story.
Readers repond: Stories and thoughts about French hugs
Boundary issues: Hugs and Shrugs