Suppose a consumer has certain quantities of two items, x and y. We’ll call these quantities x1 and y1, and the combination is written (x1,y1). Now ask the consumer, "If you had (x1,y1) in hand, but had to cut back to a lower quantity of x, say x2, how much y would it take to make you feel just as well off?" In real life, the consumer says, "Get lost! I don’t think about things like that, I have no idea what I’d do. Let the world try it and we’ll see." But for a thought experiment, suppose the consumer is willing and able to answer: "I’d need quantity y2 for that." So now we have two combinations, (x1,y1) and (x2,y2), which are equally good in the eyes of the consumer; the one with less x makes up for it by having more y. The consumer is indifferent between these two combinations.
Keep going: "All right, suppose x were cut back still further to x3. Then how much y would you need to be just as well off?" Continue this, and also extend the experiment in the other direction: "Suppose you could have more of x than x1, say x4. How much y could you get along with and feel just as well off as with (x1,y1)? Eventually, we have a large collection of combinations (x,y), all of which are of equal utility to the consumer. Draw a line through a plot of these combinations. This is an indifference curve. The consumer is indifferent among all the choices offered along this curve.
We don’t stop there, however. Now we ask, "Well, suppose you could keep x1 and have more than y1 of y, say y5. Obviously, (x1,y5) is better than (x1,y1), since you get the extra y without giving up any x. But then, it must be better than any of the points on that indifference curve which we constructed, since any one of them is as good as any other, and (x1,y1) is one of them. So now let’s repeat the original procedure, starting from (x1,y5): give up a little x, ask how much y is needed to stay just as well off. This way we draw another indifference curve, every point of which is better than any point on the original curve.
This way, we could, in our imaginations anyhow, construct an infinite set of indifference curves, showing how the consumer would rank any possible combination of x and y. Moving between the combinations on any one indifference curve is a pure substitution effect. Moving from a combination on one indifference curve to a combination on a better or worse curve depicts an income effect, generally mixed with substitution as well. Of course, we gloss over the fact that increments of x and y are usually discrete: it’s more convenient to draw continuous curves. Likewise we gloss over the fact that consumer preferences may change over time: we can imagine them to be stable long enough to talk about a couple of consecutive periods.
It is traditional to sketch indifference curves as something resembling one branch of a hyperbola. With y on the vertical axis and x horizontal, the slope of the curve gets steeper and steeper as x diminishes; conversely, the curve gets more and more shallow as y diminishes. The idea is that the less we have of x, the more y it takes to compensate us for any more reduction in x, and vice-versa.
Bear in mind that consumers don’t go surfing along their indifference curves. (If their preferences change, the whole map changes — a different set of curves go into effect.) Here we’re talking about constrained choices. When we depict movement from one combination to another on a curve, or from one curve to another, what we’re assuming is that, while the set of indifference curves remained the same, something changed about the economy’s productive opportunities, to which consumers react according to the given set of preferences..
We can represent productive opporunities with another set of curves, sometimes called "transformation curves". These are traditionally represented as hyperbola branches facing in the opposite direction. You might say they are nature’s indifference curves, showing combinations of x and y whose production out of given resources is all the same to nature. At any given moment, one such curve is imagined to be in effect; this is the economy’s income (or you can think of it as consumers’ budget). Consumers react by choosing the combination of x and y on the highest indifference curve which the transformation curve just touches. Where the two curves touch, they’re tangent, and the common slope at that point sets the relative prices of x and y.