Now that I've explained the topic of overflow, I can get to the second part of the problem: conditions. Conditions are any manner of expression that can produce different behavior when the very same instruction is executed multiple times. The most common types of conditions are equal, not equal, less than, and greater than.
The reason I put overflow and conditions under the same heading is that conditions are also based on carry overflow. If we compare two values, one of the following must be true: they are equal, the first is less than the second, or the first is greater than the second. Computers perform this comparison using subtraction, then checking for overflow. Compare unsigned 5 and 10 (in that order): 5 - 10 = -5 (a nonunsigned result) with carry. If we reverse these, 10 - 5 = 5, with no carry.
Thus, a carry indicates that the first is less than the second (this is always true, not just in these two examples). In the case of both values being equal, the result will be 0. Note that the assumption that carry indicates less than, and no carry indicates greater than, is only valid when the result is nonzero.
Signed comparisons are a bit more complicated, as it's not as simple as whether or not there is overflow. I won't go into examples of why this is (as you'd have to construct a full truth table to see the relationships), but this is how it works: excluding the case of both being equal, then the first value is less than the second if the overflow state is different than the sign of the result of the subtraction (overflow != result_sign) .