Think of your website or application like a grocery store. There are the parts you see and the parts you don’t see.
The parts you see are the shelves all stocked with products and the cashiers checking people out. That’s called the “client side” in web development.
The parts you don’t see are the manufacturing, distribution and everything else it takes to get the product to the shelf. That is what would be referred to as “server side”.
In web development, client side refers to the website interface (links, banner images, ads, etc) with which the customer interacts. The server side is all the logic that stores information in the database, completes financial transactions, etc.
There’s often crossover between the two (like the guy stocking the shelves at the store), so it can be confusing at times.