Folding Bike
A folding bike will have hinges or joints incorporated into its frame and handlebar stem. These allow it to be transformed into a more compact size. They are often allowed within areas that standard bikes are not allowed or convenient. It also means they are easier to store within vehicles.
As far as cost goes, a folding bike will usually be more expensive than a standard one. This is because they require more parts to allow the capability to fold and to be fastened securely or locked when unfolded. This more complicated design of course mean a more complicated system of manufacture. In addition to all of this, there is a small market for these designs.
There are some styles classified in the same category as a folding bike, though they do not actually involve folding. Some simply come apart into two or three parts for easier transport and storage. Generally, these designs will be referred to as "break-away," "separable," or "disassemble-able."