What are web cookies?
Web cookie is a general name for a portions of data stored by a website on user's computer for the purpose of being retrieved on subsequent visits on the same website. The data can be used for various purposes — from storing your language preferences for the website to identifying you uniquely when you return to the website. Some cookies are stored just briefly for the time of your visit, others can stay in your browser for years if not deleted.
Cookies were introduced because websites handle thousands of clients at each moment and have no way to distinguish your network connection from the multitude of other users' connections. This would make any multi-step or transactional operations impossible. So on the first connection website assigns you a random identifier (a cookie), which your browser reflects with each future connection. This way the website can distinguish your connection from the others.
The most popular cookie technology are the "regular" HTTP cookies (RFC 6265) which are stored in your browser when the server responds with a
Set-Cookie header. However, since then numerous other cookie techniques were invented, some of them quite intrusive and avoiding user control at any cost.
Because some cookies uniquely identify the user many regulatory frameworks in the world require a level of disclosure from website owners. See What is the privacy impact of web cookies? for more details.