What is the Priesthood?

"The priesthood is the eternal power and authority of God. Through the priesthood God created and governs the heavens and the earth. Through this power He redeems and exalts His children, bringing to pass “the immortality and eternal life of man” (Moses 1:39). God gives priesthood authority to worthy male members of the Church so they can act in His name for the salvation of His children." (See full article on Priesthood)

Jesus Christ used the Priesthood to heal and perform miracles

 And no man taketh this ^ahonour unto himself, but he that is ^bcalled of God, as was ^cAaron.  (Hebrews 5:4)
 

  18¶And the Lord said unto Moses, Take thee ^aJoshua the son of ^bNun, a man in whom is the ^cspirit, and ^dlay thine hand upon him;

 19And ^aset him before Eleazar the priest, and before all the congregation; and give him a charge in their sight.

 20And thou shalt ^aput some of thine ^bhonour upon him, that all the congregation of the children of Israel may be obedient.

22And Moses did as the Lord commanded him: and he took Joshua, and set him before Eleazar the priest, and before all the congregation:

 23And he ^alaid his ^bhands upon him, and gave him a ^ccharge, as the Lord commanded by the hand of Moses. (Numbers 27)

We see from these Bible verses that throughout time the priesthood has (1) been passed down by the laying on of hands [v. 18, 23] and (2) can not be taken upon oneself without this laying on of hands. Even the Savior had to be ordained or given the priesthood as we find in Hebrews 5:5-6, 10 and Acts 10:42. 

Jesus Christ ordaining the 12 Apostles

The priesthood authority of God was lost to the Earth after Christ was cricified and his apostles martyred.  For over a thousand years mankind went without this authority until our Heavenly Father saw it was time for it to be restored.  It was restored through the prophet Joseph Smith in the early 1800's and has been passed down by the laying on of hands since.  The priesthood remains in the Church of Jesus Christ of Latter Day Saints because it is Christ's church.

Conferring of the priesthood continues today

  I know that these things are true because I have prayed to Heavenly Father about it, searched the scriptures for answers, and done my best to do as He has asked me.  It was in the doing of these things that I found my witness of the truth.  Anyone can follow this same pattern to know for themselves.  Experiment on the word and seek the truth of it.