The Employees' Provident Fund is a savings scheme initiated under Employees' Provident Fund and Miscellaneous Act, 1952. The EPF scheme basically promotes the idea of savings in people and giving them assistance post retirement. It is one of the largest saving schemes available to Indian employees.
It's a part of amount deducted from the employee's salary on monthly basis and contributing to the EPF. In the end, the total amount received with some amount of interest is totally tax-free.

ESI stands for Employees State Insurance. It is a health insurance scheme provided for those who earn a monthly income of INR 21000 or less per month as wages.
It's managed by Employees State Insurance Corporation (ESIC) in which they are provided with the full medical care benefits to the insured person and his family. Medical care is also given to the retired and disabled insured persons.