The most obvious difference between the Earth and Mars is that Earth is almost double the size of its neighbour. Mars’ small size meant it cooled more rapidly and the liquid iron in its core eventually solidified, resulting in the planet’s protective magnetic field being lost. Energetic particle radiation from the Sun, the galaxy and beyond was no longer deflected and controlled by a magnetosphere. As a result Mars lost a great deal of its atmosphere, which made it even easier for radiation, particularly ultraviolet radiation, to reach the surface.
Radiation is extremely damaging to the molecules that make up living organisms but measurements made by Radiation Assessment Detector (RAD) on Curiosity rover have shown that human explorers that visit Mars and return to Earth would be unlikely to receive a fatal dose. Radiation becomes more problematic when considering any organisms that might have evolved on early Mars and the long term preservation of organic matter they might have left behind. One theory is that dormant radiation resistant microbes may still exist deeper in the martian soil and they become active if the atmosphere becomes temporarily thicker and greatly reduces the radiation impacting the surface (for example if there was a substantial release of gas from volcanism or melting carbon dioxide ice). If the European rover planned for 2018 makes it to Mars it will drill down a few metres into the soil and see if it can find any evidence of this potential microbial life. For missions like Curiosity that are searching for organic compounds in the near-surface the most promising sediments to explore will be those that have been relatively recently exposed at the surface and received a shorter dose of ionising radiation.
The most problematic type of radiation is generally galactic cosmic rays (GCRs) as they are extremely energetic and difficult to shield against, however their levels are relatively constant and easy to predict. For a mission on the order of months to a few years GCRs should not pose a major threat but for proposals like Mars One, where people will be sent with no method of coming back GCRs become a major issue. Cosmic and solar radiation is one of the biggest issues that will need to be considered for human exploration beyond the Moon.