The red components of the light are focused differently than the blue components. As a consequence, objects appear to have colored fringes around them and the image appears not as sharp, an effect which is referred to as chromatic aberration. Achromatic objectives correct this lens error for two colors of the color spectrum, apochromatic for three. Apochromatic objectives therefore produce images which are sharper and higher in brilliance. Apochromatic objectives often have the designation APO written on them.
Low cost educational and routine microscopes are equipped with achromatic objectives. This is perfectly sufficient for routine laboratory work and the slight color fringes which are still visible in some specimens may not even be noticed unless you specifically look for them.
Apochromatic objectives are significantly more expensive (with individual objectives sometimes costing more than the whole microscope) and are useful when image quality is of importance when doing photography through the microscpoe. Many amateur microscists do not own apochromatic objectives, because the gain in image quality might not be sufficient to justify their high price. A little bit of dirt on the front lens of the objective (such as a fingerprint) can already result in an image quality which is lower than the one produced by achromatic objectives, so there are other factors as well which are of importance.
Last, apochromatic objectives have a very small working distance. This means that the distance between the front lens of the objective and the sample is very small. If the specimen is too thick (eg. whole mounts of insects), then it may not even be possible to focus on the parts of the specimen which are further down without crashing the objective into the slide first.