Views is a highly flexible module to start with, but the contextual filters increase the use cases for the module by an order of magnitude. Contextual filters work similarly to regular filters, but there is one important difference. Instead of setting a filter value manually, the value is fetched from variables sent programmatically to the view. A regular filter could give you all nodes written by a specified user. A contextual filter for a node author would be able to display all nodes written by the currently viewed user, or the same user who wrote the currently viewed node. The concept is that contextual filters prepare a view for filtering, but the filter value is not yet determined. When the view is eventually called, it is also provided with data used to complete the contextual filters.
The classic example of how contextual filter values are provided to views is by the view path. If a view has the path example.com/my-view, the URL example.com/my-view/story/22 will call the view along with two values for contextual filters (in this case story and 22). But there are more ways of providing contextual filter values. These are discussed in Chapter about Page manager and Panels.
The difference between standard filters and contextual filters may seem trivial, but in practice it is huge. Contextual filters allow you to reuse views in many different situations and allow Views to interact with other parts of the website. One could write an entire book about how to use contextual filters in Views.
TIP1: Until recently, contextual filters were called arguments in Views, and much documentation and many tutorials and Views-compatible modules still use that term. If you see the term argument, it should be interpreted either as a contextual filter, or the value provided to a contextual filter.
TIP2: the standard functionality of contextual filters may be extended to accept filter ranges, such as example.com/my-view/story/21--100 by installing the Views Contextual Range Filter module.
TIP3: you can configure both an (exposed) regular filter and a contextual filter for the same field in your view, but the result may not be what you expect. For a description of the issue and a solution, see the Views Filter Harmonizer project page.