Fun with eye contact:
Find a female friend you know and do a silent, non-gesture staring contest. Remember results.
Find a male friend you know and do a silent, non-gesture staring contest. Compare.
More fun:
Try the same test, with different people, smiling, or frowning. It makes a huge difference, or my friends are just odd.
Less fun but still interesting:
With sunglasses.
One of the things that scared me when I started taking notice of my eye contact and keeping it, was that girls and women would keep their eye contact at all costs. Even now I feel most comfortable talking to them two at a time, because that way you can shift between them, not having a literal staring contest with one of them.
I would compare it this way. They look at you like they are watching a television show or play and are using your eye movements as a part of the interpretation of what you mean to them. 'Reading' a person, if you will.
That's why it's easier to to talk to girls when you are both looking at something, because you are not as aware of this fact. A tip to
girls: if someone is nervous when talking to you, it's not a bad thing, it's just the above. Almost everyone gets a little nervous when attention is on them; or else they are oblivious. Try looking at and talking about something in their hands or nearby so you both are looking at it and ask a question about it to start the conversation, and be prepared to answer a return question and to supply the followup.
For everyone: Meet their eyes when they look at you and give a little smile if appropriate, maybe nod at their points and be prepared to express your own opinion. Be able to keep relaxed eye contact when speaking points are made by the other at the very least. Glancing away when it's your turn to speak is somewhat normal as it shows you are gathering your thoughts but is by no means necessary. If you pause to gather your thoughts, a Hmm sound with the glance away shows you are about to speak. ([Hmm.] [relaxed glance] {Well, I guess it's OK.} {traversing to regain eye contact})
Note that I'm a dude, so I could be wrong I guess.