An exchange plan is a way to count calories by counting blocks of pre-counted "exchanges." So an 1800 calorie exchange plan will provide approximately 1800 calories but instead of calories, you're counting specific sized servings of each food group represented by the exchange categories.
You can think of the exchanges as "servings" of a food group (but you don't choose the serving sizes, the diet plan does. This confuses some people, because they might for example think that an apple is one fruit exchange, because it's one fruit. But most large fruits are actually two fruit servings/exchanges. Just like calorie counting, at first you have to look up the value of each food (there are books and online exchange counters - just like with calories), but over time you start to memorize the exchange values of the foods you eat most often. This is a bit more intuitive with exchange plans, because most of the exchanges follow a pattern. For example most vegetables count as one vegetable exchange fore 1 cup raw or 1/2 cup cooked. You do need to learn the exceptions (corn and potatoes for example are bread exchanges, not vegetable. Starchy beans can count as either bread, protein, or half-and-half. and rhubarb can be a vegetable - for 1 cup or a fruit - for 3 cups).
There are a variety of different exchange plans, but most (virtually all) are based on the exchange plan first developed for diabetics in the late 40's or 50's by the American Diabetes Association and the American Dietetics Association. A couple you may have heard of are Richard Simmon's Deal a Meal or Food mover (in the former you dealt yourself "cards" with the exchanges written on them, and the appropriate amounts, and as you ate the exchanges, you moved the appropriate cards from one side of the wallet to another. In the food mover, you closed little windows with pictures representing the exchanges (the protein exchange was a tiny picture of a chicken leg).
The exchanges themselves haven't changed much, which makes it nice, because almost all exchange plan cookbooks can be used by almost every other exchange plan. Sometimes there are small differences that mean you may have to do a bit of "translation," for example in the DASH diet, the exchanges are a little different - a protein exchange is actually equal to 3 standard exchanges because the protein exchange portion is based on a 3 ounces rather than 1 ounce as in standard exchanges, and DASH also includes a "nut exchange" which would equal 3 fat exchanges on other plans. This is very unusual though. Some of the very old exchange plans distinguished between veggies and leafy (lettuce, spinach, cabbage, and kale) veggies, or between veggies and dark veggies. That's an easier "translation" because in most modern exchange plans - they're all just veggies.
In most exchange plans, there are 6 exchange categories. I'll list them (and in parenthesis is the average calorie count)
Protein or Meat (55)
Fat (45 calories)
Veggie (25 calories)
Fruit (70 calories)
Milk or Dairy (90 calories)
Starch or Bread (80 calories)
How many exchanges you get from each category depends on the exchange plan you choose (I've linked some examples, below).
Exchange plans used to get quite a bit of criticism for being too high-carb, but that was a silly criticism, because that's easily fixed by trading the carby exchanges like breads and fruits for proteins and fats. That's not necessary any more because you can now find low-carb exchange plans.
There are books and online exchange counting resources, so just as with "straight" calorie counting, you can look up exchanges online or you can buy exchange list books. You can also learn to calculate exchanges by using the nutrition label (instructions are online, and also in the book Exchanges For All Occasions, 4th edition (I've read that the 5th edition does not have this information, but I don't know about the other editions).
I don't know if I'm explaining it very well, so you may find these link easier to understand
Introduction to Exchange Plans (great overview)
http://frugalabundance.com/frugal_fa...hange-plan.htm
Examples of exchange plans.
http://frugalabundance.com/frugal_fa...food-plans.htm
You can also find other helpful information on this site, so here's the index page where you can find food lists and even food journal pages (which give you room to write what you ate, but also a chart to check off each exchange as you eat them).
http://frugalabundance.com/frugal_fat_loss/index.htm