In order to get Yuffie just walk around the forested areas outside of Junon. After some battles you should run into a battle with Yuffie named "Mystery Ninja" sooner or later. Just keep trying. After you begin fight just beat on her like a normal enemy, you don't have to do anything special. After you have defeated her you will enter a dialoge sequence with her. In this sequence all the characters you where fighting with and Yuffie will appear on a grassy meadow. There is a Save Point but whatever you do DO NOT SAVE or else Yuffie will just run away and you'll have to find her again. What you do need to do, however is talk to her but be sure not to say anything stupid or overly desperate.
If you don't want to try it out the first time yourself here's how the dialogue is supposed to go:
Yuffie: You spikey-headed jerk! One more time, let's go one more time!
Cloud: Not interested.
Yuffie: You're pretty scared of me, huh!?
Cloud: .....petrified.
Yuffie: I'm really gonna leave! REALLY!
Cloud: Wait a second!
Yuffie: You want me to go with you?
Cloud: .....That's right.
Yuffie: All right! I'll go with you!
Cloud: ......Let's hurry on.
Now that you've got her a side-quest is opened when you go to Wutai