Two-dimensional Covalent Organic Frameworks (2D COFs) have attracted considerable interest because of their potential for a broad range of applications. Different combinations of the monomeric units can lead to potentially novel materials with varying physico-chemical properties. In this study, we investigate the electronic properties of various 2D COFs with square lattice topology based on a tight-binding density functional theory approach. We first classify the 2D COFs into different classes according to the degree of π-conjugation. Interestingly, this classification is recovered by using a similarity measure based on specific features of the electronic band-structure of the COFs. Further, we study the effect of aromaticity on the electronic structure of fully-conjugated COFs. Our results show that the conjugation and aromaticity are keys in the electronic band-structure of COFs.