I've only had this particular experience once, since I married the first person I dated after my first marriage ended. At the time my son was 2 1/2 years old, and the "new man" in my life lived a couple thousand miles away. We were talking for about three months, I think, before I let my son talk to him on the phone and see him over the webcam. Though I never said anything at that point about who this man was other than a friend, my son immediately started referring to him as his other daddy, so I guess it's a good thing it worked out ;). My son got to meet him in person another three months after that, the first time I met him in person as well.
The deciding factor in when to bring my son into the conversations was mostly when I knew that I wanted to keep the man in my life, and when I'd come to trust that he also wanted to stay in mine. It seems that most people do not reach that realization only three months into a relationship, so some parents may wait longer.
I think the child's age at the time you're dating also makes a big difference, since older kids are more able to understand what role this person is/might be playing in their lives, and that they may or may not be a permanent fixture. Younger kids tend to get attached quickly and won't understand if they're suddenly gone because the relationship with their parent didn't work out. That said, I think it's also very important to introduce a child of any age to a significant other quite some time before any big life-changing things happen with that person, such as you get married or they move in so that they can reach a comfort level with the person, and more fully understand how the new member fits into the family.