The discovery of the high critical temperature superconductivity in cuprates in 1986 opened a new epoch in the hunt for high-temperature superconductors and a distinct chapter of study in the field of the solid state physics. The early hopes to extrapolate to cuprates the previous knowledge on the superconductivity quickly vanished. Manufacturing high-quality samples and the high precision investigation of their physical properties by various experimental methods proved to be a must which unveiled the occurrence of unusual normal state and superconducting properties. Presently, in spite of the publication of more than a hundred thousand research papers, reviews, and monographs there is not a commonly accepted interpretation of all the physical phenomena and of the origins of the mechanism(s) resulting in the superconducting state in cuprates. Here we review those features considered to be of primary importance in the theoretical description of the superconducting phase transition in cuprates within the effective two-band Hubbard model proposed by Plakida et al. [1] and progress obtained in the study of this model based on the rigorous consideration of the symmetry properties following from the crystal structure of these systems, their spin reversal invariance, as well as from the complicated algebra of the Hubbard operators entering the model Hamiltonian [2-6].