Neediness is sort of an attitude or a personality trait which you don't necessarily want to display.
Have you ever had a girlfriend who just WON'T LEAVE YOU ALONE??? I find this happens with younger girls a lot, like with highschool girlfriends. I mean, you are dating her, you love her but she seems to just obsess over you and can't let you do anything alone.
That's pretty much what neediness is. It comes off like you are making someone your whole world to the point it becomes irritating to that person even if they like you.
When it comes to approaching women, this problem comes in early on. You get a girls number... GREAT! But now you find yourself ALWAY texting her, and constantly trying to meet her. If she's a little lukewarm on you especially she probably needs to feel like you have some value. So if you seem to constantly want to see her and appear to have nothing else going on, it's not showing much value. You are not in demand by other women. On the other hand, if she blows off your first attempt at a date and you just appear to not be phased by it and disappear, you'll often get girls coming back to you... the reason is that you're not being needy... i.e. you WANTED her, but you didn't NEED her. When you didn't get the date with her, you left her with the assumption that you weren't crying in your bowl of cheerios but instead probably out with another girl or had something else going on.
If you're already past the first date. Neediness develops into something a little worse. Even if you appeared awesome and confident initially, after 1 or 2 dates and especially after sleeping with her, people can sometimes just begin to assume they are now "yours". I'm not necessarily saying be aloof or cold but don't suddenly start acting like she is all there is in the world. I know the movies would have us believe this is how romance works but have you ever had a best friend who sometimes just seems to need to always be around you? It actually drives you apart.
I think the quote above sums it up really. She needs to feel like you WANT her (but you might have other options thus she feel like you chose her above all others) but you don't NEED her (because then it feels to her like she's your white knight and you'll have a nervous breakdown without her which isn't good!)