The term client technically refers to one computer making a request to another. The software on the computer making the request is the "client" and the software on the computer responding to the request is the "server".
For websites, since the computer making the request is your user and they're using a browser, we often refer to the browser itself as being the client. Sometimes browsers are referred to as user agents as well.
Client-side
When you visit a website, technically all the website code is hosted on their server. Some of it downloads to execute (or "run") on the browser like HTML, CSS, and JavaScript and other technologies or languages stay on the server and execute only on the server. When developers say "client side", they generally mean any code that runs in the client compared to server-side which means any code that executes on the server.
Also, sometimes the client is referred to a the "front end" and the server is referred to as the "back end". So saying "today we're working on client-side code" is the same as saying "today we're working on front-end code".