Hopefully someone can give you a more comprehensive answer, but this is a topic I studied a few years ago so can offer a simple version. If I have some more free time later I'll try and pad this answer out.
Firstly a few things to note however.
- 16th century Mexico was not made of tribes - political entities here were huge and advanced early-modern city states with high urban populations and not so much huts in the jungle type villages.
- The Aztec 'Empire' wasn't an empire in the traditional European sense. Tenochtitlan and its immediate neighbours 'conquered' most of the states of central Mexico, but didn't take direct control and install its own rulers, laws, or culture. Instead conquered peoples became tributary states through fear of Tenochtitlan's military strength.
- And finally, while the term 'Aztec' is handy for brevity, it isn't a particularly useful term. It wasn't used by the 'Aztecs' themselves and implies a uniform culture across central Mexico. The ruling peoples of the empire were the Mexica, located in and around Tenochtitlan on Lake Texcoco, a truly massive city-state which by some accounts may have been the largest urban centre in the world at the time of the Spanish Conquest.
Okay, onto your question as to why various states allied themselves with the conquistadors, or perhaps more accurately why Cortes allied with Aztec controlled states.
The city-states that aided Cortes, namely Tlaxcala and Totonac, weren't so much 'allied tribes' but subjects of the dominant Mexica peoples of Tenochtitlan. In the case of Tlaxcala, they were a smaller state that was conquered and indirectly ruled by Tenochtitlan, and so were expected to pay economic tribute and offer soldiers to aid Aztec war efforts. Also, despite them being part of the 'empire', in some sense they were still an enemy of Tenochtitlan, and routinely fought ritualistic wars against them, one purpose of which was the collecting of captives to be used as human sacrifices. Read into the Flower Wars to better understand this relationship. By helping the Spanish they were hoping to regain independence or at least erode the Mexica's authority.
Totanac was a member of the triple alliance that made up the Aztec Empire, though definitely wasn't an equal partner, and imperial power was very much concentrated at Tenochtitlan. Their hope was to destabalise Tenochtitlan and gain more power for themselves.
And lastly, prior to the arrival of the Spanish, the Empire had expanded very rapidly, reaching coast to coast in less than 100 years but other than having the biggest army, did little to solidify its authority. Conquered lands became tributary states but were allowed to keep their original leaders and there was no imposed homogeneous 'Aztec culture'. So at the arrival of Cortes the 'empire' was already looking a little precarious, and it is likely backlash against Tenochtitlan was already brewing - those with grievances against their overlords simply found they shared a common enemy with Cortes.
Obviously the topic is much more complex than this, but I hope this is a good framework in understanding the relationships between various pre-Columbian states.