An alpha source outside the body can do little harm. The alpha particles do not even penetrate the outer layer of dead skin cells on the body. But once inside the body (for example inside the lungs) they are very dangerous indeed. 

Each alpha particle causes a series of ionisations as it rips through matter. As they are so highly ionising they cause a lot of localised damage. 

E.g. A tiny speck of uranium dust contains millions of atoms (one millionth of a gram contains 2,500,000,000,000,000 atoms!). Over a long period of time these atoms decay, their alpha particles doing considerable damage in a region very close to the source. Alpha rays can only penetrate about 0.1mm into tissue, therefore a lot of localised damage occurs and the chance of a tumour developing or serious damage due to cell death is much higher than if the damage was more widely (and sparsely) spread.

Once inside the body alpha sources are very difficult to detect because the rays do not get out to the detector! You could swallow an alpha source and it would not register on a Geiger counter.

Thus alpha sources are very hazardous and are never used for medical applications. They are particularly dangerous in powder form - the danger of inhaling or ingesting them is then higher.

to look at an animated gif showing the spread of damage through the lung from an inhaled grain of Plutonium.

Beta particles are more penetrating than alpha particles. They therefore cause less localised damage. However they are still very dangerous. An outside the body source of beta particles would be able to penetrate the skin and a source inside the body would be able to penetrate about 1mm into tissue.

All nuclear radiation carries the risk of causing mutations to DNA or tumours. In high doses there is an increased risk of cell death.

Positrons do not penetrate through matter very far at all before they disintegrate on meeting up with a matter counterpart. They therefore cause very localised ionisation.

However each positron only causes a single ionisation so the damage it causess is very small in comparison to that an alpha particle causes.

The gamma rays that result from the annihilation are very penetrating therefore they do not contribute to massive localised damage either. They are therefore relsatively safe to use in medical scanning (PET scanning).

All nuclear radiation carries the risk of causing mutations to DNA or tumours. In high doses there is an increased risk of cell death.

Gamma rays hardly interact with matter at all. They are very penetrating. Therefore if the intensity is low they are used with a gamma camera for medical applications

If the intensity of the source is high gamma ray sources are very high they are dangerous inside or outside the body. Just as intense ultra violet rays can cause sunburn, they can cause radiation burns. But, these burns do not only occur on the skin's surface, they also occur within organs deep in the body, resulting in sickness and nausea and considerable pain and discomfort.

All nuclear radiation carries the risk of causing mutations to DNA or tumours. In high doses there is an increased risk of cell death.

for safety precautions when handling radioactive sources