1st Generation: To get Irish citizenship as a first generation british citizen, all you will need is your birth certificate and the birth and death certificate of the original immigrant.
2nd Generation: To get Irish citizenship as a second generation british citizen, you will need your birth certificate, your parent's birth certificate (the one where Irish descent is claimed) and the birth and death certificate of the original immigrant.
3rd Generation - Note this is not automatic or guaranteed unless your parent was registered in the FBR BEFORE you were born: To get Irish citizenship as a third generation british citizen, the main route is applying via citizen by association. In theory, it is possible. But in practice, it is highly unlikely to gain citizenship this way. You will need your birth certificate, your parent's birth certificate (the one where Irish descent is claimed), your grandparent's birth certificate (the one where Irish descent is claimed) and the birth and death certificate of the original immigrant. Along with this, a points system is in place to apply a minimum amount of points is required for the application to be considered.