To start with, let's review some basic definitions. A conclusion is a statement based on other statements in the argument. Those "other statements" are the premisses (or collectively, the premise) of the argument.
An assumption is a specific kind of logical fallacy called a fallacy of presumption. The author is presuming something to be true which is unsupported or unwarranted. Generally it comes from a gap in logic between the premise and the conclusion.
Imagine the author is crossing a river, leaping from rock to rock, trying to avoid getting wet. Each rock represents the pieces of evidence. The author hops from idea to idea, then takes a final, extra-big leap into the air (making his conclusion). The final stone below him is the assumption. If it's there, it will support the author's weight as he descends. If not, he falls into the river, and his conclusion doesn't hold. An assumption covers the inherent weakness of the conclusion. Let's look at a sample CR question:
One of the best ways you can feed your brain for better memory is by avoiding a diet high in trans-fat and saturated fat. A recent study has found that these fats, such as those from animal products, can cause inflammation as well as produce free radicals, which tend to weaken the brain's higher functions. Many meat producers, however, have pointed out that free radicals are a normal by-product of the body's natural metabolism. Though this may be true, there is still sufficient proof that a better memory can be achieved by avoiding too much red meat.
Which of the following is an assumption upon which the argument depends?
(A) Free radicals in high-enough quantities can damage or kill brain cells.
(B) Memory is considered a higher brain function.
(C) The body's metabolism does not also cause inflammation.
(D) No other fats increase inflammation and free radicals.
(E) The free radicals found in red meat are not found in other types of food products.
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The correct answer is (A). The argument's conclusion is that avoiding the fats in red meat = better memory, based on the premise that fats cause inflammation/free radicals, which weaken the brain. Despite the fact that the body naturally produces free radicals, the argument is assuming a difference in how the brain processes small versus large doses of free radicals. The conclusion only holds if too many free radicals is a bad thing. Otherwise, the author will land in the proverbial river, and his argument won't have a rock to stand on.
Remember, sometimes the conclusion is stated first, then followed by the premise; sometimes the conclusion comes after the premise; and sometimes it's sandwiched in between two sentences of premise. The ideas matter. The structure of the paragraph doesn't. That's why GMAT-gurus harp on being able to correctly identify the conclusion and the premise. If you can't pick out the moving parts, you can't begin to see how the engine works.
