As the radiation is a kind of energy, as cells absorb this energy can lead to varies kind of change in cells, as your mention in Mrs. Curie, due to the long time exposion to radiaoactive substance, her cells become cancer cell. Normal cells will terminate them after certain of time after its DNA was damaged. But cancer cells do not, as the DNA was damage, it won't kill itself but continue to grow and become functionless, and do damage to other cells, as it will take up nutritions and "swallow" other cell.
On the other case, a large dose of exposion radiation and lead to death, human body absorb too much radiation and many cells' DNA was damaged. From the normal procedure this cells will terminate them in a short period of time, but as you can imaginate, many cells kill themself in a short time will lead to disable of functions of the organ(s) from this cells, as organs are not functioning, it will make other organs got damage, finally as so many cells are dead, it will lead to death.
You can search on web of dose, it is a unit to measure exposion to radiation.