Assimilation and Accommodation
Assimilation and Accommodation are the two complementary processes of Adaptation described by Piaget, through which awareness of the outside world is internalised. Although one may predominate at any one moment, they are inseparable and exist in a dialectical relationship. The terms are also used to describe forms of knowledge in Kolb’s elaboration of the cycle of experiential learning.
In Assimilation, what is perceived in the outside world is incorporated into the internal world (note that I am not using Piagetian terminology), without changing the structure of that internal world, but potentially at the cost of "squeezing" the external perceptions to fit — hence pigeon-holing and stereotyping.
If you are familiar with databases, you can think of it this way: your mind has its database already built, with its fields and categories already defined. If it comes across new information which fits into those fields, it can assimilate it without any trouble.
In Accommodation, the internal world has to accommodate itself to the evidence with which it is confronted and thus adapt to it, which can be a more difficult and painful process. In the database analogy, it is like what happens when you try to put in information which does not fit the pre-existent fields and categories. You have to develop new ones to accommodate the new information.
In reality, both are going on at the same time, so that—just as the mower blade cuts the grass, the grass gradually blunts the blade—although most of the time we are assimilating familiar material in the world around us, nevertheless, our minds are also having to adjust to accommodate it.
Piaget was mainly concerned with children's developing understanding of the world, so for him (and for children) accommodation is no more problematic than assimilation. That does not necessarily hold as we grow older. We have ways of understanding our world which work for us, as relatively successful (i.e. surviving) adults. There is no problem in assimilating new information and ideas which fit with this world-view, but we find it increasingly difficult to accommodate to new stuff. One cognitive problem of ageing has been well labelled "hardening of the categories"!
What's "dialectical" mean? It starts with Socrates (about 470-399 BCE) whose manner of questioning (now known as Socratic and much venerated as a teaching technique) contested generally-accepted propositions in order to draw out a new understanding. It was systematised by Hegel (1770-1831), a candidate for the most obscure philosopher ever, reputed to have said on his death-bed, "No-one ever understood me; even I didn't understand me", but actually it is quite simple.
It is a notion of logical development in which an idea (thesis) is opposed by another idea (antithesis), and out of the clash emerges something else (synthesis). Its simplest and most boring manifestation is compromise, but it can lead to something really new.