System 1 and System 2 are terms used to describe two different modes of thinking. They were written about by Daniel Kahneman in the book Thinking, Fast and Slow.
System 1 is described as the "fast, automatic, intuitive approach", and System 2 is the brain's "slower, analytical mode" that can perform more complex verbal/logical reasoning.
Some simple examples of System 1 vs. System 2 thought include:
I have a dilemma where I wish I had the habit of flossing my teeth everyday, but it doesn't seem natural to me to do it. I often don't feel like doing it, or I don't remember. I know that I "should" do it because my dentist told me to, and I understand why it's a good idea—but I still often neglect to floss.
Humans aren't born with the innate desire to floss their teeth, so they have to be conditioned in some way to gain the habit. But System 1 doesn't "reason" the way System 2 does. If these Systems could talk to one another, I imagine the conversation would go like this:
System 2: I should floss!
System 1: ...
System 2: Not flossing has negative consequences.
System 1: Mehhh.
System 2: I should ward off negative future consequences.
System 1: I'm hungry.
If my System 1 isn't on board with flossing, it's unlikely I'll develop a habit of flossing. Using just System 2 to change behavior requires will power.
Better ways to convince System 1 might include feeling the disappointment of my dentist when I don't floss, looking at horrifying images of rotting mouths, or recalling the pain of my last dentist's appointment. The effectiveness of this depends on what motivates me specifically, but generally, convincing myself will be easier if I'm not just using high-level notions of what's good for me in the abstract. System 1 is bad with abstract. System 1 is bad with ideas like "many years later my oral cavity will deteriorate."
System 2 mode is where I reason out the pros and cons of flossing. Likely, I will use words to list the reasons. The System 2 mode of thought allows me to be more strategic, use numbers in my decision-making, and formulate concepts like "many years later my oral cavity will deteriorate."
System 2's domain is consciousness—everything I notice or am aware of is being processed a little bit more slowly and deliberately. Reading the words on the screen, thinking about what they mean, having an internal monologue, being aware of that vague ache in my throat (a sign I might be getting sick), noticing noises in the background, and many more things my mind brings to my conscious awareness—these are happening in System 2. I like to think of my System 2 thoughts as "what's taking up my attention right at this moment."
When I play a game that I've played a lot, many of the moves come without having to think about them. In Magic, I automatically go through the motions of drawing and playing a land. Even more complex actions can be ingrained with enough practice. This is my intuition telling me how to play. That's engaging System 1.
But sometimes I run into a tricky situation or a complex board state, and I have to pause and think using more drawn-out reasoning. I might have to consider what my opponent has in their deck or hand; I might have to figure out what the following turns will look like; or I might have to calculate damage. That's engaging System 2.
When System 2 doesn't get very involved, System 1 can cause the body to go into an "autopilot" mode. Like when you're driving a familiar route, sometimes you don't pay that much attention and still end up taking the exact route to your house.
Knowing exactly when to engage System 2 is a tricky business. Often, you're pretty happy that you don't need to bother reading street signs or noticing landmarks as you're driving—but only assuming you wanted to wind up at your house and not the gym.
Most of the things we do are "automatic." Breathing, controlling our heart rate, sweating, walking, pressing keys on a keyboard, up to more complex actions like taking a shower, eating breakfast, driving to work, and making conversation. It's when we insert System 2's guidance that we manage to change our routine and start flossing our teeth or going to the gym after work.
I would argue that distinguishing System 1 and System 2 is a fuzzy task. I can explain their different properties, but that doesn't mean necessarily that they are distinct modes—they are basically both on all the time while awake, and they are always "talking" to each other.
The way I use the words "System 1" and "System 2" on average might be even further from their intended meanings; I like to talk about System 1 and System 2 as though they were different parts of my mind that each have a unique set of wants and reasons. This often helps me talk about myself and to myself in more useful terms.
My current goal is to pay better attention to which System seems to be more "in charge" of any given situation—and whether I want to adjust the balance by engaging more of one.
The real training begins when I learn how to align System 1 and System 2. The expectation is that when I do, I'll be able to accomplish things that seem hard and effortful more easily.