Behind the website is the reservations system. If two people on that system make the same entry they get identical results. Thus, if two people ask for London/Amsterdam on a given date and there is only one seat in (say) C class on KL123 they will both see 1 in the availability. If one of the users then books the seat and the other does nothing the one who does nothing will not see any change in their display unless they also try and book the seat at which point he will get "0 available" response.
Overlaid on that is point of sale availability (which is probably irrelevant but what the heck). In this case the system will consider where the request has come from and if they are two different agencies (or airline office) they may see different responses. For instance an agent with a deal may see availability in their "special" class while the other agent with no deal will either see 0 or even not see the class at all.
Overlaid on all that is the website. Now the mainframes that the actual reservations systems use don't take kindly to being hit with lots of requests from websites - after all anyone from about age 4 to age 120 who can wield a mouse can look up flight availability via websites. So the websites use chacheing which means storing responses to provide responses to "what's available" and using the stored response instead of going to the mainframe until someone actually says "I want to book" and, of course, the cached data can be wrong.
There's a part of the answer for you. Probably not all but it's a start.