As the only two guys in my immediate circle of acquaintences who have dated single mothers, Dan and I have talked at great length about the dynamics of creating a meaningful connection when a child is part of the equation.
In a childless courtship, there is only one relationship thread whose nuance and dynamics you need to be sensitive to: the one between you and your partner (a fiendishly difficult task by itself at times!).
Adding a child to the equation means there are six relationship threads whose nuance and dynamic you need to be sensitive to (assuming, of course, you're pursuing the woman seriously, and not just as a sexual conquest):
 You and your partner
 You and her child
 Your partner and her child
 Her child and the father
 Your partner and the child's father
 You and the father
To be sure, not all of these relationships are of equal importance - but it would be reckless and irresponsible to ignore them. The level of complexity in dating single mothers are a significant leap from a relationship with a childless woman, and places moms looking for romance under a particularly challenging set of difficulties.
Single Mothers' Burden
It takes a far more emotionally aware/psychologically savvy/empathic (read: more desirable, more difficult-to-attract) man with which to forge a healthy long-term relationship for a single mom than her child-free rivals. The grim paradox from most single mothers' standpoint is that the males most capable of the necessary heightened emotional maturity/sensitivity are the hottest property in the dating market. All else being equal, most of these high-value male bachelors gravitate toward less-complex relationships, and in the process, leave single moms scrambling for whatever is left.
On top of everything else, a high degree of patience is mandatory. When a promising relationship begins, the extent of the woman's involvement with a new boyfriend must be shielded from her child, until both partners have some confidence of a possible long-term future. To do otherwise would be cruel, especially to a child who already had to endure this trauma once.
Patience. Empathy. Psychological savvy. Dan has them in spades, and when we concluded our telephone call, I told him: "Beth and her daughter may not know how wildly fortunate they are to have you - but I do. If they ever forget, I'll be sure to tell her that next time I'm in town. You take care, man."
"Heh. Thanks. You too."