The meaning of the words Rosh HaShana literally is "Head (beginning) of the year." However, this holiday carries much more significance than merely being the day on which a new calendar becomes necessary.
The Sefer HaChinuch explains that Rosh HaShana is the day on which the whole world is judged. Each individual creature is judged as an individual. This judging has been compared to sheep passing in single file under the watchful eye of the shepherd. Just as each sheep is scrutinized alone, separate from the flock, so too are we judged on Rosh HaShana, as individuals, separate from everyone else.
Why do we have a day of judgment? The Sefer HaChinuch explains further that this holiday on which we are judged is truly a kindness of Hashem. Hashem, because we have this yearly holiday, reviews our deeds yearly, thereby preventing our sins from amassing. As we "only" have to deal with our sins one year at a time, there is still room for repentance and atonement. Furthermore, as Hashem judges us with kindness, if there are few sins, they are pardoned. If there are sins for which punishment is needed to cleanse the person, the punishment is exacted in small doses, bit by bit. If the accounting of our deeds did not take place on a yearly basis, our sins would accumulate until the point that Hashem would decide to end the existence of the world, because of all the evil and disregarding of His words.
Rosh HaShana is the day which ensures the continued existence of the world. It is therefore fitting to have this day as a holiday. However, as this is the day on which we are judged, it is only proper that we conduct ourselves on Rosh HaShana with a level of fear and awe not seen on other festivals. This is the reason why we do not say "Hallel" (a special prayer of praises of Hashem): It is not fitting for a person to sing songs of praise while standing in judgment.