Communications in the terahertz band (THz) (0.1--10~THz) have been regarded as a promising technology for future 6G and beyond wireless systems, to overcome the challenges of evergrowing wireless data traffic and crowded spectrum. As the frequency increases from the microwave band to the THz band, new spectrum features pose unprecedented challenges to wireless communication system design. The molecular absorption effect is one of the new THz spectrum properties, which enlarges the path loss and noise at specific frequencies. This brings in a double-edged sword for THz wireless communication systems. On one hand, from the data rate viewpoint, molecular absorption is detrimental, since it mitigates the received signal power and degrades the channel capacity. On the other hand, it is worth noticing that for wireless security and covertness, the molecular absorption effect can be utilized to safeguard THz communications among users. In this paper, the features of the molecular absorption effect and their impact on the THz system design are analyzed under various scenarios, with the ultimate goal of providing guidelines to how better exploit this unique THz phenomenon. Specifically, since the molecular absorption greatly depends on the propagation medium, different communication scenarios consisting of various media are discussed, including terrestrial, air and space, sea surface and nano-scale communications. Furthermore, two novel molecular absorption enlightened secure and covert communication schemes are presented, where the molecular absorption effect is utilized as the key and unique feature to boost security and covertness.