This is a super old-fashioned term (no one uses this!). Think of the days when a woman's father would need to be asked permission by a gentleman to court his daughter (a gentleman in a top hat I may add), to go to the pictures and stroll down the promenade (if that's how you spell it).
It means 'dating' essentially, in quite old fashioned terms, but this whole area is very grey. I would say "dating someone" is official (like a couple), "seeing someone" is unofficial (but still going on dates, meeting up, hanging out, whatever you'd like to call it), and some would say it is casual. Courting was from a time when people typically didn't have casual relationships without the goal of marriage, kids, etc. so I would say closer to dating. It's dating with the view of a marriage partner, but it's out of common use.
But, I am in my mid-twenties so what do I know... I would happily be corrected ;)