Here is a link to the lds.org page describing the priesthood and its role:
LDS.org - Topic Definition - Priesthood
I suggest reading the "additional information" part, which has quite a bit of detail. Although that honestly covers mostly organizational stuff and not actualy responsibilities. For more on the actual responsibilities, I suggest reading
Doctrine and Covenants 20.
In short, the duty of the priesthood is to serve. By fulfilling the duties laid out in the above D&C link, priesthood members ensure that the needs of the members are being met, that the doctrines of the Gospel are being taught correctly, and that work of God is progressing on the Earth. We also have a lay priesthood - every worthy male member can hold the priesthood and participate in the ordinances for which they hold proper authority. This encourages more active participation from the "ordinary" members, and it gives the Lord a literal army of priesthood to protect and serve the Gospel and the world (LDS member service isn't limited to just LDS members).
:edit:
I should add, the duties of the priesthood include officiating in many of the rituals and ordinances we perform, which ordinances can only be performed by those with proper priesthood authority (by order of the Almighty). Priesthood authority includes things like revelation - which is how God leads the church today, sealing - the power as Christ gave to Peter that what is "Sealed" on Earth is also "Sealed" in heaven (like duplicate records are kept), and saving ordinances like baptism, sacrament, and marriage. Without proper priesthood authority, none of these things can happen and the Church starts to stray from the true path. This is what we believe happened to the early Christian church, that proper (and complete!) priesthood authority was lost and hence our belief in the need for it to be "restored" through prophetic revelation.