No specific name is given to the "Heart Virus". When one is infected with the Heart Virus, symptoms include severe malaise and fatigue. When Goku first felt the symptoms of the Heart Virus, he was out of breath just flying away from the island where the Androids appeared. As the infection progresses, the victim starts experiencing excruciating chest pains until eventually, they succumb to the virus and die. Senzu Beans will not stop this virus or even slow it down. The only way to cure this virus is with the Heart Medicine.
Future Trunks' timeline
Future Trunks mentions that many people succumbed from the virus in his alternate timeline. Unfortunately, Future Goku is one of the people who die from the Heart Virus. Six months following his death, the androids appear and murder all of the Z Fighters with the exception of Future Gohan, thus sparking the events of Dragon Ball Z: The History of Trunks as well as the Trunks Saga, Androids Saga, and Cell Saga.
In the main timeline, Future Trunks arrives from the future and gives Goku the antidote to the Heart Virus (the virus is incurable when Goku first gets it, but a cure is developed soon afterward; it was just too late for Goku). However, his presence in the past causes history as he knows it to change drastically. One major change is that Goku does not contract the heart virus when he was originally supposed to, but instead it appears several months later during his fight with Android 19 in May of Age 767. Goku is saved from death by Vegeta and taken home by Yamcha to receive the antidote. Ten days later, the virus is successfully cured, and Goku re-joins the fight against the androids and the new threat, Cell.
It is never explained why or how Goku contracted the virus six months after his alternate timeline counterpart, although a possible explanation is that Goku, knowing of the androids' arrival, was at home, training, whereas Future Goku, thinking that all was well, was out having fun with his family, where he contracted the virus. Main Timeline Goku appears to first be affected by the virus while in Sasebo (the city where the Androids first appeared), as he was noticeably exhausted after merely flying away from the city. Alternately, Goku's training may have simply postponed the virus as he was in better health.
Another possible reason is because Goku originally defeated Frieza and King Cold on Earth, not Future Trunks, therefore lessening the wear and tear on his heart. While going Super Saiyan seems to speed up the process, when not Super Saiyan the infection is very slow.
Appearances in video games
In Dragon Ball Z: Budokai, Dragon Ball Z: Budokai 2, Dragon Ball Z: Budokai 3, and Dragon Ball Z: Infinite World, it is called Viral Heart Disease and is a special capsule item that can only be used by equipping in the "custom" menu in the character selection screen. It takes 2-3 of the 7 technique slots, in game it causes both the user and opponent's health to deplete continuously, until only half of their last health bar is left. The user and opponent can avoid taking damage by equipping "Vaccine", a single slot capsule item which nullifies Viral Heart Disease's effect.