The essential answers have already been given. The basic fact is that the rainbow is a circle that is centered on the “anti-solar point” (the point on a sphere surrounding you that is the point directly opposite the sun; if you draw a line from the sun through you it hits that sphere at the anti-solar point). Since you’re generally standing on the ground, the anti-solar point is generally below the horizon, and the rainbow is an arc. The rainbow subtends an angle of about 42 degrees from the antisolar point.
There are a few interesting points:
1.) as noted above, you can see a complete circle if you’re standing on a high point and looking down into a valley, or if you’re flying on an airplane, or some other situation in which the horizon doesn’t block most of the rainbow.
2.) You can actually see a “rainbow” without being above the horizon. In some cases, round raindrops or dewdrops can be supported by the hairs on grasses. In this case a number of interesting optical phenomena can take place (look up heilingenschein, for instance), and one is that you can see the rainbow extending across the ground, below the horizon. Only, since it’;s on the ground, which intersects with that 42 degree cone to create a hyperbola, people tend to visualize its shape as hyperbolic.
3.) There is, of course, a secondary rainbow subtending an angle of about 51 degrees. It has its colors reversed (red on the inside). The secondary bow is caused by light reflecting two times from the interior surface of the drop. (tertiary and quarternary rainbows exist, but are close to the sun and virtually never seen,.)The space between the primary and secondary rainbows is darker, because light is scattered preferentially out of the area. This is called Alexander’s Dark Band after the Greek philosopher Alexander of Aphrodisias, who first described it. Anyone making jokes about Alexander’s Ragtime Band will be shot.
4.) There are two types of reflected rainbow, which aren’t centered on the anti-solar point. In one case you see you see the rainbow apparently reflected from the water’s surface – only it isn’t actually a reflection of the main rainbow itself – you’re seeing light that passes through raindrops, then reflects off the water surface – but those drops aren’t located quite where the ones making the rainbow directly are located. The point about which these rainbows form is above the horizon.
The other type of reflected rainbow is caused by the reflected image of the sun (usually from a lake or other horizontal reflecting surface) behind the observer. In this case the “antisolar” point for the reflected rainbow is above the horizon, and you may see more than half a circle. In a lot of cases, however, the raindrops aren’t present in enough of the air to see the whole thing, and you may only see a portion extending a short distance above the ground. In that case, despite its curvature, many people perceive it as straight, and it’s called a Rainbow Pillar.
There are lots of other variants, but that’s enough for now. Be aware that there are other optical phenomena that lead to color separation that aren’t rainbows, but look like them, such as the ice halos mentioned above, or sundogs, or various other ice crystal arcs.