The Enneagram is essentially a study of how we disengage from reality into our own egos, and of how we can instead learn to live in the present moment . Every personality type in the Enneagram represents a tendency toward a spectrum of behaviors, some of them healthy and some of them unhealthy. Acting from the healthy side of our type leads us to the integration of our psyches with the world. But until we learn to do so, our default is to act out the tendencies of our type, which leads us down a path of deeper and deeper disintegration.
Fortunately, the Enneagram provides us with a roadmap to understanding what we have to do, and more importantly what we have to let go of, in order to become whole and bring the sum of ourselves to every moment. Each type grows/integrates towards a certain other type, and resists/disintegrates towards another. As you can see in the Enneagram symbol, each type is connected by a line to other types in two separate sequences:
1 grows to 7 grows to 5 grows to 8 grows to 2 grows to 4 grows to 1
3 grows to 6 grows to 9 grows to 3
(The types also disintegrate in the same sequences, but in reverse order)
What this means is the each type (and their corresponding wing type) connects to two other types in a significant way. For example, healthy 2s share behaviors with healthy s and unhealthy behaviors with unhealthy 8s. But beyond that, it also means that as a 2 is growing towards emotional balance and health, they begin to "integrate" the lessons of all the other Enneagram types into their being, in the sequences laid out above.
Types 3, 6, and 9 are the "primary" types of their respective triads, meaning they inherently hold the lessons of all types of their triad. So for example, regardless of their wing type, a Type 3 deals with feelings of shame related to their outer world and how others perceive them (like 2s) as well as their inner world and how to manage their feelings (like 4s). The same can be said of 6s who deal with fear related to the external world (like 5s) and related to their internal world (like 7s), and of 9s who deal with anger related to their external world (like 8s) and internal world (like 1s).
We use the term "Integration" because as people of any base type grow, they integrate the lessons of all types of the Enneagram, hence integrating all sides of their human potential into their psyches. The because the Primary types (3, 6, 9) represent all types in their respective triads, as they grow through each other they are integrating every type as they go. Generally speaking, that means that each step forward is somewhat harder to take but they integrate more per step when they take it. For example, these types may appear to "turn a corner" very quickly, making a vast change for the better in one fell swoop. On the other hand, the Secondary types (4, 1, 7, 5, 8, 2) go through growth paths that are more stepwise and gradual as they integrate each type one by one. They can also grow rapidly, but will still do so in a sequential order.
There are many writings that expand on the process of integration towards other types, but to demonstrate it in a simple way I have assigned each type a "virtue" which is achieved when it is fully integrated. In this structure, when each type integrates a new type, they are essentially gaining the corresponding virtue. It is an over-simplified example, but I think it illustrates the process well. The virtues are:
8 Personal Power
Let's take a few examples. First we'll take a look at a Secondary type, type 8. An 8 on a growth path progressively lets go of its ego drive to control its environment in order to feel powerful and hence safe. As it does that, it follows the course: 8 - 2 - 4 - 1 - 7 - 5 - 8. The first step to let go of their ego drive is to become service-oriented rather than self-oriented (2). Next it discovers its individuality and gets in touch with its feelings (4). Next its feelings lead help it get in touch with a broader sense of integrity and principle (1). Next its principles help it find a sense of fulfillment and joy in life (7). Next that sense of joy helps it connect with the healthy pursuit of knowledge and growth of its objective capabilities in the world (5). Finally it returns to integrating its own type, 8, where it is healthy and personally empowered, bringing its fully developed gifts to others.
Now, and example of a Primary type, 9. On its growth path, 9 lets go of the fear it has of its own gut impulses (Instinctive Triad). To do that, it first grows towards 3, the Primary type of the the Feeling Triad (2, 3, 4), simultaneously getting in touch with its motivation (3) and its component parts, service (2) and individuality (4). The 9 overcomes the shame that binds it and makes it afraid to assert itself. Put another way, the 9 gets in touch with its emotions (4) and its desire to serve others (2) hence finding motivation (3). Once the Feeling Triad is integrated, 9 grows towards 6, the Primary Type of the Thinking Triad (5, 6, 7), simultaneously getting in touch with a sense of loyalty to others, feeling like a part of something more than themselves (6), and the components of that sense, feeling confident in their own capabilities (5) and feeling fulfilled in their internal world which finally allows them to connect to the world outside of themselves in a meaningful way (7). The 9 overcomes the fear that keeps it isolated from and mistrustful of others. Finally they integrate the Instinctive Triad (8, 9, 1), finding a sense of true peace in motion (9) by trusting its component parts, namely its gut impulses in the external world (8), and trusting that its internal instinctive drive isn't a bad thing (1).
While people on a path of growth and integration move through all of the types, while disintegrating they do not. This is because the process of becoming less healthy cannot go on forever, and essentially following that path will lead to psychotic break or even death before the individual disintegrates through all of the types. That said, the path of any type in the direction of disintegration is predictable, following the same sequence as the path of integration but in reverse order.
This structure in the Enneagram gives us more detailed understanding of each type individually, but more importantly it also shows us how similar we all ultimately are.