Thank you for your question. The five usul of din (tawhid, adalat, nubuwat, imamat, qiyamat) were devised my twelver theologians in order to distinguish twelver Shiism from other sects within Islam, thereby giving it an identity. These are the main concepts from a theological point of view, but religion comprises more than these 5 principles. These principles are formulated from the scriptural sources and the principles of adalat and imamat are specifically in response to other sects and theological positions that developed within the Islamic world.
The first is the unity of God. That He is unique and there is none like Him. The existence of God is proved in a variety of ways, but His uniqueness is a presupposition, even though the concept is coherent when compared to other views which lead to incoherencies. He is unique because He is unlimited and is not constructed of parts. If any other god is supposed to exist other than Him, He would neither be unique or simple (not constructed of parts) as there would be another being that would limit Him, or would be a part of Him. This position is derived from the scriptural sources that stress the uniqueness of God.
The second principle is that He is Just. His Justice is stressed as the Ashari school of theology stressed God's Power over His Justice. The Imami theologians held that it was impossible for God to punish his Prophet (saw) for example, as that would be against His Justice, but the Ashari's maintained that God could do whatever He wished and that Justice was what God said it was, not humans conceived.
Prophethood is proved through the grace of God in desiring that His creatures be guided, as well as the proofs which they brought to prove their prophethood, which could not be repeated without divine help. The Qur'an is the greatest living proof of prophethood.
Imamate is again a principle which separates Shiis from other schools of thought and for the twelver school imamate continues from prophethood and is for the same reasons, such as God's desire to guide His creatures. Imamate is necessary to safeguard and interpret the final message in every time.
The Day of Judgement is what brings all of the test of the life of this world into perspective. By it the consequences of our actions remain hidden and test has meaning. The soul is everlasting and so a hereafter must exist as a place where the soul rests after death.
May you always be successful.