Sir,
It is for your principal employer,(if the unit in which you are working is covered/coverable under Employees' State Insurance Act, 1948 and rules/regulations framed thereunder) to complete your registration under the said Act. It is compulsory, if your are coverable i.e. getting wages upto coverable limit.
If your employer is deducting ESI contribution from your wages, then ask your employer or authorised officer of your establishment i.e. HR/Time Office Incharge etc. about your Identity Card (Pehchan Card) under said Act.
Once you have Identity Card under ESI Act and eligible for benefits under said Act, it is your legal right to avail benefits as admissible as and when such eventuality and need arises.
Better you may download necessary literature/rules/regulations etc. from the website of ESIC, or you can also seek guidance by visiting nearest Branch Office of ESIC..