Hitchhiking was very common in Armenia during the 90s, but it's fallen off quite a bit in recent years. Hitching is still perfectly acceptable, though, and in more rural areas, can be especially convenient to get around and meet people. Although there is no set rule about money changing hands, it's always polite to offer something - though you're obviously under no obligation unless you've agreed to something. Most drivers will refuse to accept money.
To flag down a car, you hold your hand out into the street as cars approach, and move your hand as if you're tapping an 8 year old on the head, in a sort of "stop here" signal.