The Haftara of the first day of Rosh Hashana relates the story of Chana, who was childless and came to the Sanctuary to pray. In the merit of her prayers she was blessed with a son - the prophet Samuel. Eli the High Priest, seeing Chana so immersed in prayer and oblivious to her surroundings, suspected her of being intoxicated - not from wine, but from the very act of praying.
"I am not drunk," Chana explained. "I am pouring out my soul before the L-rd." Through prayer, Chana's soul was uniting with G-d.
On Rosh Hashana we ask G-d to fulfill our needs. Our requests are spiritual and material: We ask Him to bless us with healthy children, long lives, and an abundant livelihood.
Rosh Hashana is the day of G-d's coronation as King, as we say, "Reign over the whole world in Your glory." How do we accept G-d's sovereignty? By nullifying ourselves in His Presence. When we are completely nullified before the King, we are unaware of our personal desires, aware only of being in G-d's Presence.
This presents us with a seeming contradiction. If Rosh Hashana is characterized by an absence of self-perception, how can we simultaneously pray for the fulfillment of our personal requests?
When a Jew prays to G-d on Rosh Hashana, his prayer is an extension of the process of coronation. While superficially he may be asking G-d for material blessings, his true intention - whether consciously or subconsciously - is the desire to spread awareness of G-d's kingship in the world. By praying for material blessing, the Jew is merely asking for Divine assistance in fulfilling his G-dly mission on earth.
It was this concept that was unclear to Eli the priest. His contention was that when a Jew prays there is no room for personal requests; the awareness of being in G-d's Presence should be so intense that it precludes anything else. When Eli saw Chana praying for a child, he mistakenly concluded that she had forgotten G-d's Presence.
Not so, was Chana's reply. Her longing for a child was not a personal desire, but a wish to fulfill a greater mission in life. This is evident in the vow she made, that if G-d would bless her, the child would be given over for a life of total service of G-d. Chana wasn't asking G-d to fulfill her personal request; she was praying for G-d to fulfill His own needs!
So too is it with us on Rosh Hashana. Although our petitions are personal in focus, the true essence of our prayer is to unite with G-d.