While members of the LDS church believe in three separate and distinct members of the Godhead (God our Heavenly Father, Jesus Christ, and the Holy Ghost sometimes called the Holy Spirit), therefore making us seem polytheistic by the definition given by brandaemon, we are still monotheistic in worship.
To give a little bit more insight to resplin's comment, despite the seemingly paradoxical existence belief in the Godhead with canonized scripture, it is, in fact, in complete accordance and agreement with it. 2tim424 made a comment on Dave DeLong's post questioning how his answer works with scriptures referring to only one God.
Christ testified of the existence of the Father on multiple occasions (John 10:36; John 17:3; Matt. 6:9-13) and the Father has testified of the Son as well both at Christ's Baptism in Matt. 3:17 and on the Mount of Transfiguration in Matt. 17:5, Mark 9:7 and Luke 9:35 and the Holy Spirit is referred to as well at Christ's Baptism in Matt. 3:16 and when Christ is describing the desire of the Father to bless His children with it if they will ask Him in Luke 11:13.
Now that we have established the existence and basis for the core belief as described in the New Testament we can answer the questions about the scriptures that seem to point to only one God. When Christ was called "Good Master" by one seeking knowledge about how to gain eternal life, Christ told him, "Why callest thou me good? there is none good but one, that is, God" (Matthew 19:17 KJV) and he repeatedly counselled His disciples to pray unto the Father (Matt. 6:9, Luke 11:2; 3 Ne 13:9).
The other part of all of this is in the time period surrounding the Atonement. First, when Christ prayed to Heavenly Father the great Intercessory Prayer (John 17) He spoke with Heavenly Father in giving an accountability and report of what He had accomplished on earth. He said, "I have finished the work which thou gavest me to do."(John 17:4) indicating that His earthly mission was given to Him by the Father and part of a larger plan that the Father was in charge of. This idea that Christ answers to, and serves, God the Father is made more evident both in the Garden of Gethsemane and on the Cross at Calvary.
In the Garden, as He atoned for the sins of the world, He prayed to the Father, "Father, if thou be willing, remove this cup from me: nevertheless not my will, but thine, be done"(Luke 22:42 KJV) giving the impression that the Father was in charge of what was happening. On the cross he cried out to the Father, "My God, My God, Why has thou forsaken me?"(Matt. 27:46; Mark 15:34) as well as "Father, into thy hands I commend my spirit" (Luke 23:46) and what would logically follow, "It is finished" (John 19:30). It should be noted that the order in which He said these things is not known, and it is not confirmed that The last statement was directed to Heavenly Father, but logic lends itself to it, as it is the only account withing the gospels that doesn't specify the last words of the Savior being directed towards the Father.
Christ was very clear about His accountability to God the Father and it is a well known fact that the Holy Spirit serves God the Father as well. So, when Christ says that "there is none good but one, that is, God"(Matt 19:17 KJV) and Isaiah reads "there is no God besides me" (Isaiah 45:5) we can agree wholeheartedly, because God the Father is the Father of our Spirits and He is the One God that is spoken of. This is the extended explanation to Dave DeLong's analogy and how it is in accordance with the other scriptures pertaining to one God. As for the part where men become gods, there is a bit more explanation needed.
He, being our Father, desires the best for us, His children. According to LDS doctrine, that means perfection and so He created a plan whereby we can become like Him. This plan required the Atonement of Jesus Christ (including the Garden of Gethsemane, the Crucifixion and Resurrection) for us to be able to return to, and be perfected in, Him if we will keep His commandments and endure to the end.
Paul described our potential to become like God in Romans when He said,
"The spirit itself beareth witness with our spirit, that we are the children of God: And if children, then heirs; heirs of God, and joint-heirs with Christ; if so be that we suffer with him, that we may be also glorified together." (Romans 8:16-17)
Latter-day prophet Lorenzo Snow taught this doctrine quite powerfully as he related this experience:
"In the spring of 1840, Lorenzo Snow was in Nauvoo, Illinois, preparing to leave for a mission in England. He visited the home of his friend Henry G. Sherwood, and he asked Brother Sherwood to explain a passage of scripture. “While attentively listening to his explanation,” President Snow later recalled, “the Spirit of the Lord rested mightily upon me—the eyes of my understanding were opened, and I saw as clear as the sun at noonday, with wonder and astonishment, the pathway of God and man. I formed the following couplet which expresses the revelation, as it was shown me. …
“As man now is, God once was: As God now is, man may be.”1
Feeling that he had received “a sacred communication” that he should guard carefully, Lorenzo Snow did not teach the doctrine publicly until he knew that the Prophet Joseph Smith had taught it.2 Once he knew the doctrine was public knowledge, he testified of it frequently." (Teachings of the Church: Lorenzo Snow, Chapter 5: The Grand Destiny of the Faithful)
As for the doctrinal explanation and support of this, you can read the rest of that chapter here: Teachings of the President of the Church: Lorenzo Snow, Chapter 5: The Grand Destiny of the Faithful
This should answer most, if not all, additional questions concerning this, but if not, please feel free to ask them.