Web development is all about communication. In this case, communication between 2 parties, over the HTTP protocol:
The Server - This party is responsible for serving pages.
The Client - This party requests pages from the Server, and displays them to the user. On most cases, the client is a web browser.
The User - The user uses the Client in order to surf the web, fill in forms, watch videos online, etc.
Each side's programming, refers to code which runs at the specific machine, the server's or the client's.
The User opens his web browser (the Client).
The User browses to Google
The Client (on the behalf of the User), sends a request to Google
(the Server), for their home page.
The Server then acknowledges the request, and replies the client with some meta-data (called headers), followed by the page's source.
The Client then receives the page's source, and renders it into a human viewable website.
The User types Stack Overflow into the search bar, and presses Enter
The Client submits that data to the Server.
The Server processes that data, and replies with a page matching the search results.
The Client, once again, renders that page for the User to view.
Server-side programming, is the general name for the kinds of programs which are run on the Server.
Process user input.
Structure web applications.
Interact with permanent storage (SQL, files).
ASP.Net in C#, C++, or Visual Basic.
Nearly any language (C++, C#, Java). These were not designed specifically for the task, but are now often used for application-level web services.
Much like the server-side, Client-side programming is the name for all of the programs which are run on the Client.
Make interactive webpages.
Make stuff happen dynamically on the web page.
Interact with temporary storage, and local storage (Cookies, localStorage).
Send requests to the server, and retrieve data from it.
Provide a remote service for client-side applications, such as software registration, content delivery, or remote multi-player gaming.
Any language running on a client device that interacts with a remote service is a client-side language.
*HTML and CSS aren't really "programming languages" per-se. They are markup syntax by which the Client renders the page for the User.