Dehydration
Without gravity to keep fluids evenly distributed throughout the body, fluids congregate in the upper body. This signals to the brain that the body needs to excrete what it perceives as excess fluid. Dehydration often results.
Bone and Muscle Deterioration
The purpose of bones and muscles is to support and move the body in a weighted environment. In a zero-gravity environment, muscles and bones don't have to struggle against gravity. Over time they become weaker and are eventually useless on Earth.
Dizzy Spells
The inner ear helps humans distinguish from "up" and "down" and maintains balance. In zero gravity, there is no real up or down, and thus the inner ear has no work to do. This leads to problems once astronauts return to Earth, including dizziness and loss of balance.
Fun Fact
Those who could truly travel to the center of the Earth (and survive) would experience natural weightlessness. This is because at this point, gravity would be directed away from them in directions, leading to a net gravitational force of zero.
Misconceptions
People often equate zero gravity with being weightless, and this is not technically correct. An environment that is truly "zero gravity" does not have any gravitational force, and gravity is present in some form or another throughout the universe. What astronauts experience in space is a weightless environment, where Earth's gravitational force is gone.
Prevention/Solution
Fortunately, the effects of zero gravity are temporary and reversible. Maintaining an exercise regiment in space can combat many of the negative effects of weightlessness, including muscle and bone deterioration.