The New Testament states that Jesus rose from the dead on a Sunday around the time of the Jewish Passover. Naturally, the early church wanted to preserve chronological order by celebrating Easter after the moveable feast of Passover (first full Moon following the vernal equinox) and so determined the best way of doing so was by establishing the date of Easter as the first Sunday after the full Moon following the March equinox. Incidentally, the first full Moon of spring is also known as the Paschal Full Moon, as it is used to set the date of Easter in any given year. In addition, if the Paschal Moon occurs on a Sunday, then Easter takes place the following Sunday.
Associating Easter with the Vernal, or Spring equinox, made sense as this time represented renewal and rebirth for the Christian church. Interestingly, though, The council of Nicaea (325AD) involved the full Moon so as to provide early christian pilgrims traveling to Jerusalem, with more night sky light to facilitate a safer journey.
Unfortunately, Roman astronomy at the time was not advanced enough to realize axial precession caused the date of the equinox to change slightly every year from March 19th, 20th or 21st so the church simply fixed the date for the spring equinox at March 21st. Further complicating matters when determining the date of Easter is the fact Western Christianity now follows the Gregorian calendar, while Eastern Christianity continues to use the Julian calendar, which has accrued 13 days out of sync since 325AD. Therefore, in Western Christianity Easter falls on a Sunday between March 22nd and April 25th, while in Eastern churches it varies between April 4th and May 8th.