Apochromats are refractors whose lenses are laid out in such a way as to combine at least three colours at a common focal point. This suppresses the chromatic aberration that is typical of refractor telescopes. Without this correction, bright objects like the Moon or the planets would have disruptive colour fringes.
Refractors are telescopes where the objective consist of one or more glass lenses. The lens refracts light to a focal point. However, the exact location of the focal point depends on the wavelength (colour) of the light. This causes chromatic aberration which is noticeable as a coloured fringe around bright objects. In physics, this dependence of refraction on wavelength is called dispersion.
There is no really precise definition of the term apochromat. Most apochromats have an objective consisting of two lenses, in which a special glass is deployed. This so-called ED glass has particularly low dispersion (the abbreviation ED stands for extra-low dispersion). The ED apochromats provides a largely pure-colour image.
An objective consisting of three or four lenses will provide an even sharper image. These apochromats are called triplets or quadruplets. These can be as good as mirror optics in term of colour purity.