How poison kills you depends on the kind of poison. The one thing all poisons have in common is that they will interfere with one or more biochemical pathways necessary for life. Most, but not all, poisons are neurotransmitters that interfere with the way your nerve cells function. Chemical weapons such as Sarin gas is one example. Other poisons can be swallowed or absorbed through the skin.
For example, eating poisonous mushrooms or drinking water contaminated with salmonella or E. coli can kill you if not treated quickly. These substances can enter your bloodstream and cause kidney and other organ failure.