I will endeavor to answer these questions as simply as I possibly can. Just remember, it is a rather complex subject. ;)
What is the Priesthood?
The Priesthood is the Authority to act in the name of God.
"And there were great and marvelous works wrought by the disciples of Jesus, insomuch that they did heal the sick, and raise the dead, and cause the lame to walk, and the blind to receive their sight, and the deaf to hear; and all manner of miracles did they work among the children of men; and in nothing did they work miracles save it were in the name of Jesus."
-- 4 Nephi 1:5
What is the Priesthood used for?
The Priesthood is used to serve and bless the lives of others. It cannot be employed for the sole benefit of the one who holds the Priesthood. Christ used the priesthood to Heal, Feed Thousands, Kill and Destroy, as well as to Teach others, such as when He Walked on Water.
The Priesthood of God has been there from the Beginning. It is the power by which God created the Heavens and the Earth. (Colossians 1:16) Even as God is from Everlasting to Everlasting, so also is His Priesthood. (Psalms 90:2)
"all things continue as they were from the beginning of the creation."
-- 2 Peter 3:4
Where is the Priesthood valid?
Because God is omnipresent, His Authority is also omnipresent. It does not matter where someone who holds the priesthood is at, it is not the priesthood of man, it is the Priesthood of God.
How do you get this Priesthood Authority?
A man may receive the Priesthood by what is called the Laying on of Hands. After receiving the Priesthood, most ordinances are also performed by the Laying on of Hands.
"Then laid they their hands on them, and they received the Holy Ghost.
The Priesthood can not be bought and sold as a thing of the world, because it is not of this world. For a man to have and use the Priesthood, he must have his heart right with God.
"That the rights of the priesthood are inseparably connected with the powers of heaven, and that the powers of heaven cannot be controlled nor handled only upon the principles of righteousness.
"And he laid his hands upon him, and gave him a charge, as the Lord commanded by the hand of Moses."
-- Numbers 27:23
Who has the Priesthood?
"And no man taketh this honour unto himself, but he that is called of God, as was Aaron."
-- Hebrews 5:4, (see above photo)
The Priesthood is for those to whom it is given, and they are to use it to bless the lives of others. No one can simply say that they have the priesthood and expect God to recognize the ordinances they perform.
For example, if you are driving down the road and some guy in an ice cream truck flips on that annoying yankee doodle song with some flashing lights trying to pull you over. Well maybe there is something important that he needs to tell you, so you pull over.
He then tries to give you a speeding ticket. You were speeding, and you did pull over for him.
Are you going to pay that speeding ticket? NO.
Now rather a Highway Patrol car starts up his siren and lights. You pull over because you were speeding. The office then writes you a ticket.
Are you going to pay that speeding ticket? YES.
This is because the Officer had the Authority of whatever state he was employed in to enforce the law.
So even with good intentions, if a Baptism is performed by someone without the Proper Priesthood Authority, that Baptism is not recognized by God any more than a speeding ticket from the ice cream guy.
Why is Priesthood Authority so Important?
When you do not have the Priesthood Authority to do something that requires it, you end up in a bad situation. This often means immediate consequences whether you see the connection or not.
"Then certain of the vagabond Jews, exorcists, took upon them to call over them which had evil spirits the name of the Lord Jesus, saying, We adjure you by Jesus whom Paul preacheth.
When did the Restitution of all things happen???????
The Aaronic Priesthood was restored to the earth by John the baptist on May 15, 1829. The Melchizedek Priesthood was restored through Peter, James, and John shortly thereafter.
"Then he called his twelve disciples together, and gave them power and authority over all devils, and to cure diseases.
Scriptures on Priesthood Authority : Mark 6:5, 4 Nephi 1:5, Colossians 1:16, Psalms 90:2, 2 Peter 3:4, Acts 8:17-21, Doctrine and Covenants 121:36-38, Numbers 27:23, Hebrews 5:4, Acts 19:13-16, Ephesians 4:11-14