When you perform a ritual, or if a priest performs a ritual on your behalf, a special sankalpa mantra will be chanted in your name along with your specific prayer. This sankalpa mantra acts as a bridge between you and the energy and auspiciousness created for you by the ritual. Through the sankalpa mantra, the effect and benefits of the ritual will be directly transferred to you.
References
http://www.nithyananda.org/events/rudra-homa-rudra-abhisheka#gsc.tab=0