We can learn something about the interior structure of the giant planets by studying a spacecraft's response to the planet's gravitational field. A large sphere with a dense core will exert a different gravitational pull on a spacecraft than a sphere with a uniform mass distribution. Based on results from previous spacecraft visits to the outer planets, we believe that all four of the gas giants have a small rocky core at their centers, surrounded by a layer of icy material.
The interiors of Jupiter and Saturn are so dense that the hydrogen gas gets compressed to very high pressures, eventually so that the hydrogen behaves like a metal. This material is very unusual and is very difficult to reproduce in a laboratory on Earth because it is difficult to achieve such high pressures.

Image courtesy of Tristan Guillot.
Jupiter's magnetosphere is the largest structure in the entire solar system! It is even larger than the Sun. Remember, a planet's magnetosphere is defined as the region around the planet within which the planet's own magnetic field dominates over the general interplanetary magnetic field. Magnetic fields in a planet are caused by a) metallic material and b) currents inside the planet. Earth has both of these, so does Jupiter. Venus does not. Here is a comparison of the relative sizes of magnetospheres in the solar system.
Here are some good illustrations of Jupiter's magnetosphere.
The presence of a magnetic field on the giant planets can cause aurora, just like we observe on Earth.





In 1995, the Galileo Probe entered Jupiter's atmosphere, providing us with our most close-up sample of giant planet atmospheres. It measured the structure of Jupiter's atmosphere, and detected Jupiter's stratosphere and troposphere.

Probe results. Image from NASA/Ames.
Convection is the process responsible for weather on Jupiter, just as it is for the Earth's atmosphere. Gases from below rise, cool, and condense to form clouds. Jupiter's atmosphere has three kinds of clouds:

Image from The New Solar System, Beatty & Chaikin.
The temperature profile of each giant planet (shown above) dictates where the clouds will form - different molecules condense at different temperatures. Saturn contains the same cloud layers as Jupiter, but they are more spread out due to Saturn's lower gravity. Uranus and Neptune are cold enough to form methane clouds in their tropospheres. Methane absorbs red light and reflects blue light, hence the bluish appearance of Uranus and Neptune.
Jupiter and Saturn are probably colored reddish/brownish/yellow due to the interaction of sunlight and the gases in the atmosphere - it creates a "smoggy" compound (like what sits over Los Angeles or Juarez). However, the detailed nature of the colors is still not very well understood!
The giant planets all have fast east-west winds. The cause of the alternating jets is not well-understood, but it is likely related to the rapid rotation of the giant planets. Large storms also exist on these large planets: for example, Jupiter's Great Red Spot and Neptune's Great Dark Spot. Jupiter's Great Red Spot is a vortex (like a cyclone) that is maintained by Jupiter's high east-west winds. Why don't hurricanes last for hundreds of years on Earth?

