Asymmetry and dissymmetry have two different meanings in chemistry. I've seen some dictionaries which list technical definitions along the lines of the one Pieter quotes: "symmetry, but in opposite directions as the two hands". I'm not sure what this definition means, or if it makes sense as a technical definition, but it does sound like a mangled version of the correct definition for the term as chemists use it.
In chemistry, the term 'chiral' (stemming from the greek word for hand) describes the property of not overlapping with one's mirror image. Left and right hands are chiral because they are mirror images of each other, but however you reorient them, you will not be able to make them overlap. Conversely an object such as a chair can be reoriented such that it is indistinguishable from its mirror image, and therefore is not chiral. Every asymmetrical object, or an object which lacks any elements of symmetry, is chiral, but not vice-versa. Dissymmetrical objects lack a particular element of symmetry called an 'axis of improper rotation', but they may or may not have other elements of symmetry. Every chiral object is dissymmetrical (asymmetrical objects are dissymetrical by definition), but not vice-versa. A chair happens to be dissymmetrical, but is not chiral. A screw is an example of a dissymetrical object which is not asymmetrical (screws have rotational symmetry) and is chiral (right and left handed screws are non-overlapable mirror images).
I'm sorry if I went into too much detail. My goal was to explain the relationship between the various technical terms, which the dictionary definition does not reflect.