Space is a harsh environment for humans—we didn't evolve to live there. Systems throughout our body have specifically evolved to function under the stresses of gravity.
Because our muscles are constantly working against gravity on Earth, those muscles have a lot less work to do in space. In other words, they'll start to atrophy. That includes our heart (which shrinks a little), the muscles in our neck that hold up our heads, and our calves.
Since our bones don't experience the same loads they do on the ground, they feel no need to maintain themselves. Bone tissue is still absorbed, but it doesn't rebuild itself like it would on Earth. This absorbed bone can cause kidney stones, and the places where the bone has weakened can easily fracture.
Here's a graphic that shows the rate of bone loss for various parts of the body:
Aside from the deleterious affects of microgravity, a potentially larger threat is radiation. We really do take for granted how much our planet protects us from it.
Out in space, astronauts are exposed to galactic cosmic radiation, solar radiation, solar cosmic particles, and geomagnetically bound radiation. All these types of radiation are so prevalent that, when astronauts close their eyes, they often still "see" bright flashes, as cosmic rays hit their optic nerves.
Radiation exposure can result in cell death or cause mutations resulting in cancers or epigenetic effects. Some systems are quite sensitive to radiation and their functions can quickly become impaired, such as the immune system, bone marrow system and eyes, which could develop cataracts.
We do our best to provide shielding, but paradoxically, shielding can result in more exposure by what is called secondary radiation. When an energetic particle hits the shielding material it can emit gamma rays and energetic neutrons. NASA takes astronaut radiation exposure very seriously. We both monitor the exposure of each crew member for individual missions and maintain a total career tally. When they hit the allowed threshold, they can be prohibited from flying ever again.
source: gizmodo.com by Robert Frost