A region is x-simple if every line parallel to the x-axis satisfies one of:
(1) The line does not touch the region.
(2) The line touches the border of the region, either at a single point or along an unbroken line segment.
(3) The line passes through the interior of the region, once.
Another way of saying this is that for any two points on the line that are in the region, all the points in between are in the region too.
The terms y-simple and z-simple have similar definitions, for the y-axis and z-axis respectively.
Regions which are x-simple are relatively easy to integrate over with x as the dependent variable. In a 2-D system, then, the interval of integration (on x) will be from f(y) to g(y), two functions of y, and y would be the independent variable (meaning the interval of integration will be from some fixed value (the lowest bound for values of y for any point in the region) to some other fixed value (the highest bound for values of y for any point in the region)). The way this works, just considering the integration on x, is: y specifies which line parallel to the x-axis, f specifies the least value of x in the region on that line, and g specifies the greatest value of x in the region on that line. The values of f and g need not be defined for lines that don't pass through the region at all, because they'll only be evaluated at values of y that actually do pass through the region; this is ensured by the interval of y integration.
In a 3-D system, the interval of integration will be from f(y, z) to g(y, z), two functions of y and z. Again, these specify the least and greatest values for x in the region on the line specified by the values for y and z. The problem remains, then, to integrate over y and z. Project the desired region onto the y-z plane; this gives a 2-D region. Construct the y and z integrations around that 2-D region, and you've solved the problem.
In most textbook exercises, a 2-D region to be integrated over will be y-simple (and most of those that are not y-simple are x-simple). Also, most 3-D regions will be z-simple, and the projection of the region onto the x-y plane will be y-simple or x-simple (but some will be y-simple, with the projection on the x-z plane x-simple or z-simple, and some will be x-simple, with the projection on the y-z plane y-simple or z-simple).
Some regions are not x-simple or y-simple (or z-simple, if the problem is in 3-D). The simplest approach is to split the region up into subregions, each of which is simple for some axis. Integrate for each subregion, add the results, this gives the integral for the whole region. Another approach would be to change the coordinate system, e.g. a ring (2 ≤ x2 + y2 ≤ 3) can be integrated easier using polar coordinates, because it is r-simple. This won't always work, though; at least, not without a lot of work.