Fascinating question, and I'll answer it in a two-pronged fashion

1. Depending on the species of the anglerfish, the diet of the male anglerfish will dramatically change, but can be grouped into two broad categories: prey to be captured/eaten and passive nutrition from a female, which I'll discuss more in #2.

2. For certain species of anglerfish, in particular ceratioid anglerfish, the males cannot eat without first finding a female anglerfish. Some males of these species have highly developed olfaction (ability to smell) and eyesight to locate females. This superb sense of olfaction is used to locate females by sensing the strong pheromones/attracting odors released by the females. After finding a female, a male bites into the skin, releases enzymes that digest tissues in both the male and female, which then mediates a fusing of the two and sharing of a circulatory blood system. Upon this, the male becomes nourished ("eats") when the female captures prey, digests, and the nutrients enter the shared bloodstream of the male/female pair.

As far as I can find, most male anglerfish don't really "eat". In all species, the males have the primary goal of finding females to bite and then fertilize. In many species, the males fuse with the females, becoming essentially parasites which extract nutrients from the blood of the host female. After this, most of the male (fins, eyes, some organs) gradually falls away as these pieces are unnecessary. In species where the males do not fuse with the female, I did not find much information on the males eating. Even in such species, the males are not predatory and therefore lack a lure, large mouth, and teeth. Since many (female) anglerfish eat dead matter that has sunk from the ocean above, males of some species may also eat this type of food. [Though I cannot guarantee that these males eat anything at all; they may just swim around until they die.]

Great question Christina. Anglerfish are super strange and interesting creatures! Male anglerfish have come up with a strategy to ensure they get to mate with a female. He bites her side and then attaches himself there indefinitely! The skin of the female starts to grow around the male and eventually they share a circulatory system. The tiny male is kept alive by what the female eats, and is ready to fertilize eggs whenever the female spawns.

Current research shows that male anglerfish act as parasites and physically attach themselves to females. The male then receives its nutrients from the female's bloodstream. Some research also suggests that at least for certain species of anglerfish, before the male finds a female to whom to attach, its diet is not significantly different from that of the female - so mainly fish and some molluscs.

Here are some interesting links:
ScienceLine.

Here is a picture:
picture.

I am not sure, but prior to attaching to a female, I would guess that they eat zooplankton, based on their size. Once they attach to a female, they become parasitic on the female they attach to.

Anglerfish is a fish from the order Lophiiformes. Male symbiosis only exists in the suborder Ceratioidei. The male ceratioids look for a female and then attach to it permanently. Scientists believe this is due to the difficult of finding a mate in the deep sea. The digestive tract in those male ceratioids do not properly develop. This prevents them from feeding.

There are many anglerfish species with free living males. They are about the same size of their female counterpart and eat same kinds of prey.