The process of Borg Assimilation is possibly the most invasive and profound mutilation any humanoid, or indeed, non-humanoid lifeform can suffer.
What makes assimilation so terrifying is the speed with which the Borg can carry out the procedure. A single injection from a drone's tubules can carry as many as five million nanobots, which proceed with the assimilation at a characteristically brutally efficient pace.
The first organism in the body to be attacked is the victim's blood, or in the case of bloodless species such as the Breen, the victim's airways. The nanobots immediately alter the molecular structure of the red blood cells to carry fuel and electrical impulses as opposed to Oxygen, or whatever gas the victim needs for survival, whilst simultaneously harvesting metallic elements such as iron from red-blooded species, cobalt from blue-blooded species etc. for use in creating more nanobots. This alteration causes the victim to start to suffocate, as Oxygen is deprived from the vital organs of the body, and also helps to stifle any resistance the victim may attempt.
With the body being starved of Oxygen, the nanobots head straight for the brain stem to proceed with phase two of the assimilation. Micro-generators are hastily constructed out of the metallic elements, as well as bone and nervous tissue at the base of the skull, and key areas of the brain are "rebuilt" to accept Borg energy as opposed to Oxygen. At the same time, the interplexing beacon is constructed inside the drone's cerebral cortex, and becomes functional within minutes in most drones, allowing them to link to the hive mind. Often, extraneous parts of the brain are left to wither and perish, the space often being filled by extra processors or generators.
The damage to the brain is reversible using modern-day methods- full recovery is likely provided the drone had not been assimilated for more than five days, and disconnection and removal of 85 of the implants can be accomplished amongst "life" drones.
With the victim subdued and the brain undergoing its transformation, the nanobots proceed toward the victim's other vital organs. What you have heard about a drone's heart stopping is entirely true- in most cases, the heart is converted into a manufacture/repair plant for nanobots, and will eventually stop its compressions when the outer walls become reinforced. This lack of heartbeat, and therefore lack of Oxygen to the remaining outer layers of flesh and skin, is what gives Borg drones their characteristic deathly pallor. It is also the reason why drones are all hairless, and why they can survive and operate in a vacuum. "Bionic zombies" is a colourful, yet accurate term that is often applied to the Borg. Often, the liver and the stomach will be transformed into power plants and/or specialised operating areas. Coiled intestines and/or other similarly-packed organs will often be left to atrophy by the Borg, or converted into the tubes you often see running across a drone's surface.
As drones neither eat, drink, excrete nor procreate, the rectum and genitals are often sealed and/or removed entirely and replaced by either speciliased operating equipment or extra power/nanobot plants. The external body armour covering a drone from its neck down is grafted directly onto the flesh itself- removal and regeneration of the often fully-decayed skin is typically the most time-consuming part of "unassimilation".
A drone's bones also receive internal and external reinforcement, particularly the skull. Very often limbs are fully amputated and replaced by specialist equipment that is tied directly into the drone's central processor, however in 95 of all drones, at least one hand is left relatively implant-free, due to the efficiency of the design of the opposable thumb. Species that lack this characteristic, such as Yridians, are usually the recipients of particularly specialised implants.
The implants often found on or inside a Borg's head and face are also wired directly through the skull and into the central processor. Ocular implants are usually implanted directly into a drone's retinas, and then wired straight through the optic nerve and into the brain itself. Very often, there isn't enough left of the eye to salvage during "unassimilation", and an artificial replacement must be constructed in its place. The bones of the ear are also subject to the same treatment, often destroyed and replaced entirely by implants.
Overall, a drone is designed for speed and efficiency over aesthetics. However, the pain and discomfort endured by a drone, not to mental the deep psychological trauma, is intense to say the least. In the case of the Borg, pain truly is irrelevant.
Dr. Joseph Zimmerman, lecturing at Exbiology 211, "Borg Assimilation", Starfleet Academy, San Francisco, Earth, Stardate 81516, February 16th, 2404