Both the NICEIC and ELECSA are government appointed organisations. Their role is to make sure that all member electricians are performing their work to the necessary standards and issuing certificates where required. Annual checks are also made by ELECSA or the NICEIC on previous jobs to make sure they comply with current BS7671 electrical regulations and the electrician is also checked for competence and the presence of liability insurance.
Both companies have close links and are essentially the same thing.
Any notifiable work carried out by a registered electrician is logged with either ELECSA or the NICEIC and that information passed on the local building control (part of your local council) and the customer.