1. Approach the person whose phone number you want. If you have met them before, use their name as you get close. "Hi Isabel! How have you been?" If you don't know them, ask someone you know for an introduction.
2. Start a conversation with him or her. This works whether or not you have already been introduced. Don't try to be witty or impressive, just be yourself.
3. Keep it warm, friendly, but brief. You don't want to monopolize their time.
4. Wind up the conversation by saying the following: "Hey, I don't want to hold you up. Can I call you sometime so we can talk more?".
5. Keep moving! If the person urges you to stay and talk longer, then do so at their request. Don't end the conversation, get their number, and then stand around.
6. Call them in a few days, even if you just leave a message saying "Hi, I just wanted to say how nice it was to see you the other day, and I'm looking forward to seeing you again. I'll try you again, or please feel free to call me."
7. Make sure you leave your name and phone number.
8. Keep smiling in the event of a rejection, and say "OK, well here, I'll give you mine. I'd love to hear from you again."
9. Give them your number and name written neatly on something noticeable like a brightly-colored napkin. Don't use a torn scrap of receipt from your wallet.