I want to know if it's possible to control a refrigerator compressor using Arduino?
the regular relay will do the job?
if not what should I have?
At least you need sensors, actuators and an Arduino.
Depends on the compressor. Post the details.
To answer your question Yes! Will the regular relay do the job, no idea, the lack of information on the compressor makes this impossible to answer properly. Same for if not what should I have. You have the start of a very interesting question but more details are needed and what is it going to do. Does the compressor need a starting relay or are you going to fake it, assuming single phase. Add the additional information and you will probably get several good answers.