Traditional hosting is a general term referring to non-cloud-based solutions that have been around a long time. These are still very appealing solutions in a variety of situations. It's most cost-efficient to start as simple as possible and grow your infrastructure over time.
Shared hosting is cheap, but can get the job done in a wide variety of cases. The main premise of shared hosting is exactly what it sounds like – you share a computer with a variety of other users. There are more strict limits to system resources as well as overall system control. This is fine when beginning and for simple projects. You will want to move away from a shared host after your project reaches a certain point of complexity.
Get the control and independence of a dedicated machine for a fraction of the cost. With a VPS, a server is subdivided into multiple virtual computers that appear as dedicated computers to the user. This is a great technology with plenty of room to upgrade as your project grows.
The main differentiator between “traditional” hosting and “cloud” hosting is the idea of “on demand” resources. With traditional hosting you pay a fixed monthly price for a fixed amount of resources. With cloud hosting solutions, resources are typically on-demand. You pay as you go, with a much more streamlined path to scalability.