At first, Christians didn't have an exact way to determine Easter's date. The Bible mentions that Jesus' death and resurrection occurred at the same time of year as Passover, so people would simply celebrate it around then, usually on the first full moon after the vernal equinox. But, since the Bible didn't state an exact date, Easter ended up being celebrated on different days around the world.
In 325 CE, the Council of Nicea decided that all Christians needed to observe Easter on the same day. So it was agreed that Easter Sunday would always be the first Sunday after the full moon that follows the vernal equinox.
For example, let's look at this year. The vernal equinox is tomorrow 20th March, and the next full moon is 31st March. If we follow the formula that was laid out way back when, this year's Easter Sunday should be Sunday 1st April — and, in fact, that's when it is.
Of course, this is only Easter's date within Western forms of Christianity, such as Roman Catholicism and Protestantism. Eastern Christianity uses the same formula, but follows the Julian calendar, as opposed to the Gregorian calendar. Usually, this leads to Easter being celebrated a little later in the Eastern Church (although this year is an exception, since Easter falls on April 16 for those following both Eastern and Western forms of Christianity).
Luckily, if you ever forget the trick to knowing Easter's date, the Astronomical Society of South Australia has calculated the holiday's date for the next 200 years. That should have you more than covered.