Alpha particles, beta particles and gamma rays are all ionising radiation. They can knock electrons off atoms and molecules, turning them into ions. When this happens inside living cells it can damage them.
A low dose can damage the DNA in a cell and cause mutations. If a mutated cell then divides out of control it can form a tumour, so ionising radiation can cause cancer, often years after the exposure. A high dose can kill cells outright, which causes tissue damage such as radiation burns and radiation sickness. The risk depends on the type of radiation, the size of the dose and the time of exposure.
Because of these dangers, precautions are taken. Sources are kept in lead-lined boxes and handled with tongs, never by hand. This keeps people at a distance and shielding reduces the radiation reaching them. People also limit the time they spend near a source and never point a source at anyone.