Handymax is a term for a class of cargo ship. Unlike the similar terms Suezmax and Panamax, Handymax does not refer to a size constraint tied to a particular navigation requirement but is more of a general class. There is no firm specification (as in engineering) that I've been able to locate; it seems more a category used by business to describe or classify their vessels.
A Handymax ship generally was between 30,000 and 50,000 DWT. 50,000 to 60,000 DWT are referred to as Supramax class, but modern Handymax ships seem to be blurring the lines at between 52,000 and 58,000 DWT. In addition, they are between 150 and 200 meters in length; however, there is a noticeable trend to remain under 190 meters as there are one or more loading terminals in Japan which have limits of about that size.
In addition to the size, a Handymax ship comes with five separate holds, and its own cranes for loading and unloading at ports with less available infrastructure. There are a few liquid carriers - tankers - referred to as Handymax, but the majority are dry cargo. They do not carry standard containers (TEUs); rather, they carry bulk - ore, grain, dry chemicals and the like. Since they do not carry TEUs they are easily recognized; a house or superstructure near the rear of the vessel, and three or four large cranes spaced along the centerline of the ship mounted between the five holds which are arranged longitudinally down the ship's length. The cranes are typically rated at at least 30 tonnes lifting capacity each, and run off ship's power.