This article on Cisco's website is the clearest explantion of omnidirectional antennas I've found.
What isn't so clear to me is...
1) Why is the dipole antenna the most common design for omnidirectional antennas?
2) For example, why not have an antenna that has a power source on one end, and then emits current to the other end?
3) Also, why does the radiation pattern (given on the linked page) take an odd spherical-like shape, instead of say, a cylinder?
The dipole antenna is the most common because it is inherently simple and very inexpensive to produce.
Your example, doesn't make much sense, beause that is exactly what is done with all antennas. You have a power source (the transmitter) on one end. The signal is radiated based on the shape of the antenna and any characteristics of the antenna to focus the signal.
As to why a dipole emits an "odd spherical-like shape" (often referenced as a toroid or doughnut), that is a relatively simple concept. A dipole antenna is often just a straight piece of wire. The amount of surface area available directly relates to how much signal is transmitted in any direction. You have a much larger surface area around the lenght of the wire than you do at the tip or base, allowing much more signal to be radiated from the sides.
Q1: A simple dipole antenna emits radiation most strongly perpendicular to the axis of the antenna. It emits very little out the ends. The antenna has no power variation in the circular direction around it's "waist." (This is simply the nature of electro-magnetism.) This radiation pattern matches the most common usage pattern -- the various other receivers/transmitters are situated at about the same height when the antenna is oriented vertically. When you factor in that dipoles are easy to construct (they are physically simple structures and electrical systems), they become an obvious choice.
Q2: What you suggest is exactly a dipole antenna. Note that a textbook dipole antenna is TWO wires, in a straight line, with the transmitter/receiver connected to the near ends of the wires. If you just have one wire, with a trans/rec connected to one end, you still have (in term of the E&M theory) a dipole antenna -- it's less efficient, but even easier to construct.
(Note that I'm really glossing over the E&M theory here. In reality, a trans/rec has two connections by definition. Whatever is in the trans/rec has two electrical sides. For best efficiency, you connect them to the two sides of a dipole antenna. You can leave one side disconnected -- but that's just a really small second side for the E&M purposes, giving you an unbalanced antenna. An example of a really cheap dipole setup was the older car antennas: Straight wire sticking up, with the radio receiver connected to the wire and the other side to the car's ground. So the whole metal car was the other side of the antenna.)
Q3: That cisco page has information on different types of antennas. An omni directional antenna would be as you described: The distance from the antenna for a given intensity would be a circle. (The pattern is called a Toroid.) The diagram you're asking about shows a varying distance from the antenna for a given intensity -- that pattern is often said to have lobes. The advantage of that antenna is that it is better in particular directions; Notably, it's primary lobe is significantly better than it's other lobes, making it work much better in one direction, thus is is more or less unidirectional.