The general definition of assimilation is the process of two different things coming together to blend and, in some cases, create a new thing all together. However, that is a very simplistic understanding of the process itself as there are many different types of assimilation.
The following are some examples of assimilation:
Another example of religious assimilation would be that of the Romans and Greeks. When Rome conquered Greece, they adopted their gods; Zeus, ruler of the gods, became Jupiter, Poseidon, god of the sea, became Neptune, Hades became Pluto, and the list goes on.
For example, in the North East of America, there is a very distinctive, almost nasal, accent. Within that large group, there are smaller pockets of accents -- a Boston accent would be different from a Brooklyn accent, for instance -- however, both accents are easily identifiable as the larger whole of a “North-Eastern” accent. Linguistic assimilation also occurs in popular slang. Often, neighboring communities will have different slang words for the same things. If you move from one place to another, over time, you will quite often pick up on, and assimilate your own language with, the new slang or dialect.
These examples of assimilation should give you a better idea of all of the different types of assimilation and how they occur.