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The phrase "Armor of God" is derived from Ephesians 6:11: "Put on the full armor of God so that you can take your stand against the devil's schemes" (New International Version). As a Biblical reference, the metaphor may refer to physical armor worn by God in metaphorical battles, or it may refer to vigilant righteousness in general as bestowed by the grace of God (Romans 13:12, New International Version): "The night is nearly over; the day is almost here. So let us put aside the deeds of darkness and put on the armor of light." 
The full quote as outlined in the King James Bible, is from Apostle Paul's letter to the Ephesians 6:10-18: (10) Finally, my brethren, be strong in the Lord, and in the power of his might. (11) Put on the whole armor of God, that ye may be able to stand against the wiles of the devil. (12) For we wrestle not against flesh and blood, but against principalities, against powers, against the rulers of the darkness of this world, against spiritual wickedness in high places. (13) Wherefore take unto you the whole armor of God, that ye may be able to withstand in the evil day, and having done all, to stand. (14) Stand therefore, having your loins girt about with truth, and having on the breastplate of righteousness; (15) And your feet shod with the preparation of the gospel of peace; (16) Above all, taking the shield of faith, wherewith ye shall be able to quench all the fiery darts of the wicked. (17) And take the helmet of salvation, and the sword of the Spirit, which is the word of God: (18) Praying always with all prayer and supplication in the Spirit, and watching thereunto with all perseverance and supplication for all saints; 
Other related passages, by Apostle Paul and references by later theologians, indicate a metaphorical context for the Armor of God. For instance, Paul's letter to the Romans indicates not a literal, but a figurative, application of the concept (Romans 13:12-14, New International Version): (12) The night is nearly over; the day is almost here. So let us put aside the deeds of darkness and put on the armor of light. (13) Let us behave decently, as in the daytime, not in orgies and drunkenness, not in sexual immorality and debauchery, not in dissension and jealousy. (14) Rather, clothe yourselves with the Lord Jesus Christ, and do not think about how to gratify the desires of the sinful nature. 
In terms of the parts of the Armor of God, the various pieces (the belt of truth, the breastplate of righteousness, the "shoes" of the gospel of peace, the shield of faith, the helmet of salvation, and the sword of the Spirit) are correlated to what Paul would have witnessed firsthand as the arms and armor of Roman legionaries during his life in the Roman Empire.
In Biblical exegesis, the vast majority of Biblical scholars, Catholic, Protestant, and otherwise, agree that Paul the Apostle used the concept as a reference to spiritual battle with the Devil. Christians are to put on this armor and deal well with its upkeeping. This armor seems to be in direct correlation of that of the Roman Empire's soldiers.
In the Biblical text of Ephesians chapter 6, there are 6 pieces of armor: the helmet, shield, loins, feet, sword, and breastplate. These pieces have a description of what they are: helmet of salvation, shield of faith, loins girt with truth (belt of truth), feet shod with the preparation of the gospel of peace (peace), the sword of the spirit/word of God, and the breastplate of righteousness.