The field effect transistor based on carbon nanotubes (CNT) is a very promising candidate for post-CMOS microelectronics. Transport in the CNT channel is dominated by the Schottky barriers existing at the metal source contacts. The nature of the metal and the geometry of the contact appear to influence strongly the electrical behavior, but the mechanism is still rather obscure. Extensive calculations based on density functional theory performed for both end and side contacts and for two metals of very different nature, namely, Al and Pd, allow us to identify a clear connection between the character of the chemical bonding and the height of the Schottky barrier (SBH). Our results emphasize that a low SBH for hole conduction in a CNT implies that the pi-electron system of the latter is almost exclusively involved in the chemical bonding with the metal atoms at the interface and that the bonding is not too strong so that both orbital hybridization and topology are preserved. This is the case for Pd in both end and side configurations and to a large extent for Al but in the side geometry only. On the other hand, the coupling of the metal states with the sigma-like system or, in other words, the perturbation of the conjugation of the pi-system via sp(3) C-hybridization is the mechanism that enhances the SBH. This is especially evident in the end contact with Al. By showing how the chemistry at the interfaces determines the SBH, our findings open the possibility of better controlling and designing "good contacts'.