First of all let's understand what a client and server is.
A Client is something that is used to make requests through the web, it can be a machine or program(like browser) and Server is basically a machine that provides some services like providing files and webpages. It is something that takes Request and then it Responds to the requests.
So a Client-Server Architecture is just a web architecture that classifies computers into client and servers.
It is based on centralized structure.
Let's take it further to understand the picture of How Web Works
The picture above shown is request-response cycle which uses the client-server architecture with the help of HTTP protocol.
HTTP = HyperText Transfer Protocol is a messenger of the web that is an application layer protocol that allows web based applications to communicate and exchange data. It can be used to deliver things like photos, videos and documents.
So if two computers(client and server) wants to communicate or exchange data usually in the form of request-response cycle, so both of them must speak HTTP communication protocol. In this client is the one who makes the request and server is the one who responds to the request made by the client.
Bonus :
There is another way to communicate via decentralized structure that is peer-to-peer model..yes you know about it if you have downloaded a software or movie via BitTorrent. So in this model one is a Seeder which has the data or file and the other one is peer who downloads the data via web.....and yes without the use of web servers.