Today Ne’ilah is confined to Yom Kippur, but in Temple times it was said during the year too.
The name is a shortened form of ne’ilat sh’arim, “the closing of the gates”.
The Jerusalem Talmud (Ber. 4:1, 7c) records a difference of opinion as to its meaning. Rabbi Yochanan said it was the closing of the Temple gates; Rav saw it as the closing of the gates of Heaven. Maimonides (Hil’chot T’fillah 1:7) followed the second view. In both cases it indicated the end of the day.
The idea is not that the gates of Heaven close at a particular time after which no prayer is heard. Countless statements confirm that God listens to heartfelt prayer no matter when it is offered.
The purpose of Ne’ilah is rather to ensure that we do not leave our final appeal for mercy to the very last minute of the day.