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Courting tends to mean "The Chase". It's when you are expressing an interest in a girl/guy ( depending on your preference ). You are trying to get this person to be interested in you.
Dating is when you literally set a date to see each other. The courtship is over, you are now at the "getting to know each other stage". It is too early for sex at this stage.
Next stage is "going steady" ( in UK ) where you have an established relationship, which may or may not involve sex.
After this comes engagement and marriage.
Hope this helps. It's all based on UK. Same in other countries, but names may be different.1THIS IS NOT RELEVANT ANYMORE