There are two priesthoods in the LDS church.
The lesser priesthood is called the Aaronic priesthood, the higher one is called the Melchizedek priesthood. There are different offices in each priesthood. The Aaronic has Deacon, Teacher, and Priest. The Melchizedek has Elder, Seventy, and High Priest. There are other callings within each priesthood, but those are the major divisions.
If the person is raised in the church, the male is usually ordained to the Aaronic priesthood at age 12 and is called to the office of Deacon. At 14 he becomes a Teacher, at 16 a Priest. When he is called on a church mission, or the Bishop (local parish leader) thinks he’s ready, the male is ordained to the Melchizedek priesthood and is called to be an Elder.
Females are not called to the priesthood in the LDS church, therefore have no authority to give blessings or officiate in church meetings.
What it means, according to the doctrine, is the person has the authority to act in the name of God. Depending on the office, the person can officiate in sacramental (Eucharistic) duties, baptize, confirm, give blessings to the sick and other blessings as required, and ordain others to the Priesthood. Church leaders such as Bishop, counselors, Stake President, General Authorities (the worldwide leaders) are required to be Melchizedek priesthood holders.
And my take on the FLDS and Fundamental groups (I know Monty’s take is different, I know the main body of the LDS Church has a different view, but this is MY opinion, take it for what it’s worth!) is that they are LDS. Yes, they split off from the main body of the church. But they use the Book of Mormon, their doctrines are very similar, in fact they claim to use the original church organization and doctrine that the founder Joseph Smith organized. And the main body of the church has changed a lot of what Smith taught, calling it revelation from God.
Standard Disclaimer: I was raised in the LDS church, but am no longer a member, having resigned from the organization. I do know church history and organization quite well.