Coaxial cable (coax) is by far the most popular type of transmission line used in amateur radio. Its advantages are numerous. A good quality coax is reasonably immune to RF interference from outside sources and usually does not cause RF interference to other systems. It's unaffected by nearby metal objects, such as towers, antenna masts or conduit. Properly installed, it lasts for years in any kind of weather. Many types of coax are flexible and easy to install. Also, it is easily available with a characteristic impedance that closely matches that of typical amateur radios, making impedance matching devices unneeded. Most coax cables are either 50 or 75 ohms, with the 50 ohm coax the most common for ham use.
It's called "coaxial" because that's the way it's constructed. It has a center conductor (D above) surrounded by an insulator (called the "dielectric") (C) which in turn is covered by a metallic braid or foil (called the "shield") (B). That is finally covered with a plastic outer jacket (A) for protection against the elements and physical damage. Thus, all components share the same axis.
Coax is an "unbalanced" transmission line, meaning that one of its conductors is grounded; i.e., the shield. So the signal on each conductor is not the same. As such, the shield performs two functions; it protects the cable from RF interference and is one of the two needed signal conductors.
Another type of coax is often called "hardline" because it is much more stiff and hard to bend. This is because its shield is a solid copper or aluminum tube. Often the tube is corrugated, to make bending a little easier. The advantages of hardline include its near total invulnerability to RF interference. It also has the lowest loss of any coax type. However, it's substantially more expensive, and so is not used often for ham installations.
Ladder line, also called "window line" is a type of open-wire, twin-lead transmission line. It consists of two parallel conductors separated by an insulating material. This material has rectangular openings cut into it, hence the "ladder" or "window" reference. This lightens the line and reduces the amount of surface on which dirt and moisture can accumulate, making ladder line less vulnerable to weather-induced changes in its characteristic impedance.
It is "balanced" in that the signal on each conductor is the same. The characteristic impedance of regular solid twin-lead is typically 300 ohms, while ladder line is commonly specified as 450 ohms. Some types of ladder line is much larger, however, consisting of two parallel wires with widely spaced (6 inches plus) plastic or ceramic insulating bars and having a characteristic impedance of 600 ohms or more. This type is more usually called "open wire line".
The main advantage of ladder line is its exceptionally low loss for signals in the HF band of frequencies. This can be important, since HF antennas are often large, requiring them to be installed some distance from the ham shack.
Ladder line is not used for the higher frequencies. Disadvantages are that it's more fragile than coax, cannot be installed inside conduit or near metal objects, and experiences a significant attenuation increase when wet. These conditions greatly complicate ladder line's installation. Another issue is that a "balanced-to-unbalanced" conversion must be made, at least at the radio end, and often on both ends. An impedance conversion is also usually required, to change the 450 ohms of the cable to the 50 ohms expected by the radio. Fortunately, "baluns" (for "balanced-unbalanced") are easily available for this chore.
Here is an image of some of the more popular types of connectors used on coaxial cable: