To bio-accumulate literally means to accumulate in a biological system. However, it is commonly taken to measure the uptake over time of toxic substances that can stay in a biological system.
Everything in a biological system has a biological half-life, that is a measure of how long it will stay in that system until it is lost, excreted, degrades, reacted into something different, etc. Most substances have a short half-life, as they are metabolized, or excreted as waste.
However, some compounds may stay in a system for a much longer period of time, for example calcium in the human body is laid down in bones and teeth, and even when bone cells die, their calcium is used again in the building of bones. This is sensible re-use from the point of view of efficiency, and use of scare resources.
The problem arises when toxic substances stay in the body for a long period of time. These substances are not acutely poisonous, otherwise they would kill straight away. These substances are associated with chronic poisoning.
If the input of a toxic substance to an organism is greater than the rate at which the substance is lost from the organism, it is bio-accumulating that substance. Thus, the longer the biological half-life of the substance the greater the risk of chronic poisoning, even if environmental levels of the toxin are very low.
This is one reason why chronic poisoning is a common aspect of environmental health in the workplace. As people spend so much time, for so many years in these environments, very low levels of toxins can be lethal over time.
An example of poisoning in the workplace can be seen from the phrase "as mad as a hatter". The process for stiffening the felt used in making hats involved mercury, which forms organic species such as methyl mercury, which is lipid soluble, and tends to accumulate in the brain.
Other lipid (fat) soluble poisons include ethyl lead compounds (the lead in leaded petrol or gas in the US).