Start by looking at distortion. Certain types of lenses distort photographs, some more than others. You often see greater distortion in ultra wide-angle lenses. A fish-eye lens, for example, captures a wide-angle but the difference between them and a traditional wide-angle lens is that the latter fixes the distortion. Better quality lenses have much less distortion thanks to a number of elements inside the equipment. Of course photography software does exist that allows you to correct this problem, but wouldn't you rather skip the problem to begin with?
Next comes speed. The lens is responsible for capturing and focusing light. The speed of the lens refers to aperture, the maximum amount of light that the lens will let in. A faster lens has a larger aperture and therefore lets in more light. A slower lens has a smaller aperture and lets in less light. Aperture size is measured in an f-stop. And just to make things a bit confusing, the larger the aperture, the smaller the f-stop number. If you're looking to capture fast-paced events, such as sports or wildlife, it is crucial to have a faster lens. It is also better if you take photographs in low-lighting because you can keep the ISO down, decreasing noise in the picture.
Digital camera lenses of poorer construction will have issues with vignetting. Vignetting is when the edges of the photograph are darker than the rest. This happens when more light enters the center of the lens than the edges because the lens was not constructed properly.
Better quality lenses also take care of two other problems: aberration, both chromatic and spherical. Chromatic aberration is when the lens fails to focus all colors to the same convergence point. This results in discolored edges of objects. Spherical is distortion caused by the spherical lens shape and when light hits the edge of the lens instead of the center. The result? Problems with resolution and clarity of the photograph. Neither are wanted, but thanks to awesome technology in quality digital camera lenses, this shouldn't be a problem.
Yes, high quality digital camera lenses are going to cost more but you always get what you pay for. With more money spent on better brands, you can capture much better results.