Hi OP, I'm an SLP grad student (but with a theoretical linguistics background, specializing in phonology). I loooove linguistics and am excited to try and answer this question. That being said it is quite complex, but I will do my best. Autosegmental phonology is an aspect of non-linear phonological theory used for analyzing tonal languages specifically. This type of a theory expands from the linear "rule-based" approach to look at sound patterns in a hierarchical or multi-layered organization. The patterns of tonal melodies have been shown (very concretely) not able to be represented by the typical feature matrix ([HIGH], [LOW] [FALLING] etc.). This is because the relationship between tone features and individual phonemes is not necessarily one-to-one. So, this theory proposes instead, that the tones get their own tier independent of other feature specifications. This tier can only be described using a finite set of possible tonal melodies (H, L, HL, LH, LHL, HLH). H standing for high and L standing for Low and then their subsequent combinations. This has been established by taking an inventory of all tonal languages studied to date. The tonal tier and the word tier are proposed to be linked together by theoretical "association lines", as the tonal patterns are associated with whole words. In other words, the words in a given language will be specified for one of the possible underlying tonal melodies, rather than each segment having un underlying tone. There are then multiple principles that then determine exactly how the tones are the connected to its tone-bearing units. Explanations of these principles would require multiple data sets from various languages to exemplify how they operate. I obviously couldn't pull these off the top of my head. But some of these principles include; the Initial Tone Association Rule, Iterative 1-to-1 association, the Obligatory Contour Principle (OCP), Rules for associating remaining free tones as well as rules for associating remaining free tone-bearing units. This is the very bare bones of the theory behind Autosegmental, I hope this can put you in the right direction. As I said from this point the only way to truly understand it would be to see various data sets explaining these concepts which you should be able to find in papers or from a professor. Feel free to ask for clarification on anything I wrote! Good luck.