There isn't really a modern term, because the thing itself doesn't exist any more.
When it existed originally, it was the time when a man sought to persuade a woman that he was worthy of her attentions, with the hope that ultimately she would accept a proposal of marriage.
Henry courted Mary throughout the summer.
Then, perhaps after the First World War, it became used for the time when couples were spending time together in what was perceived by all to be a relationship that would lead to a proposal and marriage.
Arthur and Agnes are courting. It won't be long before he pops the question.
By the late 1960s it was becoming more common for couples to enjoy a romantic relationship without necessarily thinking of marriage, and very soon after that it became less socially unacceptable for couples to live togetherwithout being married.
Nowadays we have such expressions as:
I am seeing Mabel = I have some sort of relationship with her that is more than just normal friendship.
I am going out with George. This is perhaps a little farther along the road to romance, but not necessarily.
Pete and Mary are an item = Their relationship is well established.
There are many other expressions, but none, I feel, that really means the same as court.
I should add that I am well past the age of 'going out' with anybody. Younger members may be able to contribute more up-to-date expressions to describe their frivolities.
