Note:
       you are meeting a random celebrity randomly, not at a planned event like a meet-and-greet or convention.
Nonchalantly approach him or her. Stay composed. Don't start freaking out and start screaming. You might scare the person away. Just because they're celebrities doesn't mean they enjoy hearing fans scream in their face. Try not to act too suspicious either.
Try to make eye contact and smile. Doing this will give you confidence and you will be able to hold your nerves. It will acknowledge your presence and let them know, "Hey! I'm here and interested!" Remember to smile; it will make you seem friendly and approachable. Try not to smile so long that it creeps them out, though.

Wait for your cue. If s/he acknowledges it with a similar nonverbal gesture, it's your cue to say, "Hey there. How are you?" or something along those lines. If s/he ignores you, give it up and walk away. S/he probably does not want to be bothered. If they reply, start your conversation.
Talk about regular stuff. Try to talk about ordinary everyday things, things you would talk about to any other person. Ensure that the conversation has a significant purpose and you're not just chitchatting. Don't bother asking them about the weather or something boring.

Keep it brief. They don't have all the time in the world, and as much as they love their fans, they'd most likely rather not hear your life story. Talk for a couple minutes and then end the conversation. You may want to think of a backup thing to say before you ever meet a celebrity, therefore you won't panic, and will already know what to say. If you really idolize this person, keep the conversation short, but never boring. Try to make it something they remember, considering they probably see hundreds of other fans a day.

Make sure that you close the conversation, not her/him. Don't talk for a long time if she/he doesn't seem interested. Example: "It was really nice talking to you, but I have to go. Would you mind signing this little sheet of paper for me?"
Remember to thank them. If she/he obliges, say "Thank you" and walk away with a little wave.
Don't get mad if they won't give your their autograph. If she/he refuses, say "Oh well, that's okay. I still enjoyed meeting you. Have a nice day." A graceful ending is always best.

Don't ask for a phone number or address. They will most likely deny. They may feel like it is an invasion of privacy, and it will make the situation feel awkward.

Tips
Be polite and respectful.
If s/he compliments you, say "thank you." Don't be brash or conceited about yourself, because they don't know us either.
Treat him like you would treat any other person you meet in public. We are all human.
Don't talk like a journalist. Talk as if you are talking to your friend. They don't want to be interrogated. If you're a journalist, do not write about your casual, nonchalant conversation to the public. It's not your job to transcribe random meet-ups.
Remember: as much as we know their names and a bunch of other trivial information about them, we do not know them! Don't expect him to act upon your own expectations and impression of him, judging only from public information which may not be accurate at all.
Some people don't like talking about their work unless it's something to be proud of, so be generic in your conversation.
Get to know him or her, but don't be a stalker. Try and talk to them about their career. Ask general questions like "How did you become such a professional?" or tell them how much you like her/him.