The pilot flame doesn't stay on. The moment I release the button the pilot goes off.

You either need to replace the thermocouple or the (thermopile) millivolt generator depending on which your unit uses.
A thermocouple has a small diameter probe, about 3/16" in diameter and 2" long, that is mounted within the pilot flame and has a very small copper tube, 1/8" diameter, that is connected to the main valve.
The probe of a millivolt generator is larger, Approximately 3/8" in diameter and about 2" long, and has a cable with two wires that connect to the main valve. Usually one red wire and the other white.

Michelle, That means the thermocouple is probably bad.

How to Replace Water Heaters Thermocouple

https://www.google.ca/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=0ahUKEwiIzcaHquzKAhWhvoMKHX5gCuQQyCkIHTAA&url=https%3A%2F%2Fwww.youtube.com%2Fwatch%3Fv%3DjDvo4yJ1OdE&usg=AFQjCNHuKU9nETvft6qM92bu0PfqzLZTBw