Basically, courting is a lot like dating, but it’s more like a pre-engagement. Basically, dating with the intention of marriage, or for the discernment of marriage.
A lot of courting is a lot more family-based then dating. Say, instead of going out to the movies w/ your girl, you go to her house and play games with her parents. Or go play cards with the grandparents. You involve the families more–they get to know you, you get to know them, etc.
Courting is more of a…permanent…thing than dating, typically. You would court someone if you think [are almost positive] that you would like to spend the rest of your life with someone, and just need to figure out if it is right. Dating, typically, is more for companionship for awhile…at least in the high school sense.
What I see of it, courting is the more old-fashioned form of dating. Instead of using each other for your current good, you help each other to your future good…chastity is very much expected. As is respect, and most importantly, love.