All of our manned spaceflights except for Apollo have been in low Earth orbit. Nestled deep down inside of Earth's magnetosphere protects you from a lot of radiation. Being above the ozone layer means that you are exposed to more ultraviolet light, but this isn't a big deal because it's not like you "go outside" much, you'll always have a window or a helmet between you and the Sun and those will always have a UV filter. The real issue is particle radiation, which comes from the Sun, the galaxy, and from parts of Earth's magnetosphere (the van allen belts). In low Earth orbit a lot of the solar and cosmic particle radiation is blocked, and as long as you steer clear of the van allen belts at high altitudes you'll be alright. The aluminum skin of spacecraft will protect you from some radiation, and you can add additional mass, especially water, to add more radiation protection. Overall in Earth orbit the radiation environment is fairly mild though a bit on the concerning side if you were to live there for decades.
If you have to travel through the van allen belts or outside Earth's protective magnetospheric bubble then things get much more difficult. In the case of the Apollo program they were careful to ensure that the thickness of the spacecraft's skin was enough to block a goodly amount of particle radiation and they designed the mission profile such that the spacecraft passed through the van allen belts as quickly as possible (spending as little time exposed to their radiation as could be managed). And in terms of operations in orbit around or on the Moon the main advantages was again one of time. Astronauts only spent at most a few days up there, limiting their radiation doses. However, they were extremely vulnerable to solar flares, which could have subjected them to extremely high doses of radiation that they had no way to protect against. In that they just got lucky.
A Mars mission, however, will spend months in interplanetary space and the chance of the crew being subjected to the radiation from a solar flare are very much higher. Additionally, they'll be subjected to much higher accumulated doses of radiation just by being in interplanetary space for so long. The thin aluminum skin of a spacecraft won't be sufficient protection for these conditions, so they'll need both a greater level of day to day protection and also an area of heavy protection that they can use to ride out solar flares and CMEs. Generally large masses of water or organic material work the best in protecting from the high energy protons that are the most concerning source of radiation in interplanetary space.
P.S. There's another option for shielding which is to generate a miniature magnetosphere around a spacecraft. This has a lot of advantages but the biggest disadvantage is that it would require additional equipment (weight is the biggest constraint on any Mars mission) as well as a significant amount of power and the technology hasn't been proven yet. Here's a neat paper on the subject: http://www.ess.washington.edu/Space/M2P2/rad.shielding.pdf