Simple and direct:
* "I'd like a French manicure, please."
* "I'm getting a French manicure."
More descriptive:
* "I'd like a French manicure with a [color] tip and a [color] base."
* "Can I get a French manicure? I want a classic look."
Adding information about shape/length/finish:
* "I'd like a French manicure with a square shape."
* "I want a French manicure with shorter nails."
* "I'm looking for a French manicure with a gel finish."
* "Can I get a French manicure with a matte top coat?"
If you have a specific design in mind (e.g., thin tip, deep smile line):
* "I'd like a French manicure, but I want the white tip to be very thin."
* "I want a French manicure with a deeper smile line."
* "I have a picture of the French manicure I want. Could you do this?" (And then show the picture).
Pro-Tip: It's always a good idea to be as specific as possible to ensure you get exactly what you want. Don't hesitate to ask questions about the process or show examples of the style you desire.