A web server is a lot like a clerk at a busy deli full of pushy customers. The customers at the deli shout requests: "I want a half pound of corned beef!" and "Give me a pound of pastrami, sliced thin!" The clerk scurries around slicing and wrapping to satisfy the requests. Web clients electronically shout requests ("Give me /catalog/yak.php!" or "Here's a form submission for you!"), and the server, with the PHP interpreter's help, electronically scurries around constructing responses to satisfy the requests.
The clerk has an advantage that the web server doesn't, though: a memory. She naturally ties together all the requests that come from a particular customer. The PHP interpreter and the web server can't do that without some extra steps. That's where cookies come in.
A cookie identifies a particular web client to the web server and to the PHP interpreter. Each time a web client makes a request, it sends the cookie along with the request. The interpreter reads the cookie and figures out that a particular request is coming from the same web client that made previous requests, which were accompanied by the same cookie.
If deli customers were faced with a memory-deprived clerk, they'd have to adopt the same strategy. Their requests for service would look like this:
"I'm customer 56 and I want a half-pound of corned beef." "I'm customer 29 and I want three knishes." "I'm customer 56 and I want two pounds of pastrami." "I'm customer ...