Gas and Ice Giants
I. Formation

When the solar system formed, it contained so much angular momentum that it needed to create a disk of orbiting gas. This would have been primarily hydrogen, but hydrogen does not stick together until you have so much of it that it has a strong gravity (such as what holds a star together), so it is expected that what first happens in the orbiting disk is that refractory molecules (those that stick together without gravity) will create dust, which amalgamates into asteroids, which develop enough gravity to pull in any molecule moving slow enough to be trapped. Far enough from the Sun, volatile molecules (especially H2O and CO2) are cold enough to form ices, which can stick to the growing rocks, and add additional mass. At distances large enough from the Sun, even hydrogen is moving slow enough to be pulled in by the growing gravity of these rocky and icy planets. If a significant amount of hydrogen is pulled in, we have an ice giant (like Neptune, which contains a large amount of hydrogen gas), and if so much hydrogen is pulled in that the whole planet is mostly hydrogen, we have a gas giant (like Jupiter).

As discussed earlier, the thermal speed of gases is proportional to the square root of T/m, and the escape speed from the planet is proportional to the square root of M/R of the planet, so we are essentially comparing T/m to M/R in order to decide if a particular gas can be contained in the atmosphere. Of course for a gas giant, the "atmosphere" at some point becomes the planet itself, but only if it can be trapped initially, so this is why gas giants (and their smaller cousins, the ice giants) are found only at large distances from the star, beyond the "frost line" where ices form.
II. Rapid rotation
Since the large mass of the gas giants requires pulling in gas over a large volume of the orbiting disk, it is natural to assume that the gas giant will form with a large angular momentum, and hence be rapidly rotating. This creates a strong coriolis effect, leading to strong winds on both Jupiter and Saturn. Since the Sun heats these planets more at their equator than poles, and warmer gas rises, this induces convection patterns called "Hadley cells" that interact with the coriolis effect to induce strong westward and eastward winds, in stripes called belts and zones. The same thing happens on Earth, creating the trade winds and the jet stream, though the Earth's rotation speed is much less and so these winds are not as extreme on Earth. On Jupiter, the winds produce clearly visible stripes, and also the famous "Great Red Spot." A similar spot is also seen on Neptune.

The cause of these weather patterns is similar to a key factor in Earth weather-- the presence of regions of low and high pressure. If a planet did not rotate, then it would not be possible to sustain high and low pressure regions, as gas would simply flow out of the highs and into the lows. Yet we know from watching weather forecasts here on Earth that high and low pressure centers last for days, and that is because the Earth is rotating. The rotation is fast enough that Earth's equator moves at 0.5 km/s, which is obviously much faster than any wind we encounter, so the air moves with the surface. But when the air moves north or south, it encounters a surface that is not moving at the same speed any more, so the air begins to either lead, or lag, the surface below. You can easily see that in the northern hemisphere, air that moves northward will lead the ground, and air that moves southward will lag the ground, so attempts to fill in a low pressure region will produce air movements that rotate around the low in a counterclockwise fashion. This is reversed for air trying to move out of a high pressure region, and both are reversed in the southern hemisphere. In the extreme case of a strong low pressure center, this counterclockwise movement of the air can generate winds in excess of 100 mph, and of course that is a hurricane. If the Earth were not rotating, not only would we never have hurricanes, we would never even have sustained high and low pressure centers moving around-- and weather would be drastically simplified (and easier to predict!).

On Jupiter, the winds are much faster (well over 200 mph) because the rotation is faster, and high and low pressure regions can be sustained for very long times. The Great Red Spot on Jupiter is a counterclockwise rotation in the southern hemisphere, so this is a high pressure center, and it has lasted possibly as long as when Galileo first pointed a telescope at it. It has certainly been there for centuries, largely unchanged, though it seems possible it could at some point break up. The spot is pinched between oppositely moving stripes, so between a belt and a zone, which feed into the counterclockwise rotation. Again these opposite motions in the atmosphere are due to the action of the coriolis force on gas being swept either to the north or south by the Hadley cells.

III. Rings
Gas giants can have their own orbiting disks of material, which look like rings. The rings are not orbiting gas, they are orbiting chunks of ice and rock. In the case of Saturn's famous rings, the chucks range from primarily baseball to bowling-ball size, and are densely enough arranged such that an astronaut could effectively "swim" through them. They also exhibit empty circles within the rings, called Kirkwood gaps, caused when the orbital period is commensurate with the orbit of a large moon. For example, the famous "Cassini division" in Saturn's rings is caused by a two-to-one resonance with the moon Mimas, whereby every time a ring particle would go around twice, Mimas would go around once, so be in exactly the same place to produce exactly the same tiny tug. Accumulating many such tiny tugs, all exactly in phase over time, would eventually pull the ring particle out of that orbit, leaving the gap. The same effect is seen in the Kirkwood gaps in the main asteroid belt, though in that case the gap is well disguised by the fact that the orbits are elliptical, so it is their semi-major axis, not instanteneous distance to the Sun, that reveals their orbital period.

IV. Moons
Gas and ice giants have strong enough gravity that they can trap large moons, and also would have formed moons with the planet as a kind of reservoir of angular momentum, much like mini solar systems. But if the moon ventures too close to the planet, tidal stresses can tear it apart, which is believed to be the source of the rings. The rings are temporary, because the orbits of the chunks will eventually decay due interaction between the particles and tidal effects, so the reason that Saturn has strong rings must be that it more recently destroyed one of its moons than for the other giant planets. Similarly, Neptune's weak ring is likely due to a moon that was destroyed longer ago, and Jupiter must not have destroyed a moon for a long time (but it seems likely that Io will eventually create an amazing ring for Jupiter). The rings exhibit remarkable many-body gravitational effects, such as the previously mentioned gaps, and also moons that "shepherd" ring particles into particular orbits (so the gravity of the moons both gives and takes away possible orbits), and moons whose orbits swap with each other at the point of their closest approach. Some moons, most notably Saturn's moon Titan, can have strong atmospheres, though at that distance it is so cold that only N2 is still gaseous. The orbits of all these moons are highly circular and tidally locked to the planet, but the fact that tiny Io is still volcanically active (while our own Moon is not) shows that sometimes the moons perturb each others' orbits, causing tidal deformations that can heat the moons' interiors before the orbits again circularize and tidally lock (at which point there is no longer any tidally induced heating).