What type of laser eye procedures are pain-free and good for person with sensitive eyes?

Not that simple: I assume you are talking about laser procedures to correct your vision? There are many laser vision correction procedures, lasik/lasek/prk/intra-lasik etc. Which one is best for you is dependent on the physical attributes of your eyes, your profession, hobbies and visual goals. See a fellowship trained refractive surgeon to determine what option can give you the best vision at the lowest risk.

None: There is no laser eye procedure that is completely pain-free. For vision correction, lasik generally is least associated with pain, versus prk. At procedure time, you may be offered a relaxant if you are eye sensitive. Light sensitivity may be more after the procedure but usually for few days only and then back to your normal sensitivity.

Lasik : Lasik is a painless procedure and would be your best option as long as you are a good candidate otherwise.