What happens when more than one user inserts data in Database (MySQL, Postgres) at exactly same time? How does it prioritize which record to be inserted first and which one later. If the answer is specific to application of program, I am asking in reference to web-applications.
It's never exactly the same time. One will happen before the other. Which one will, unless you implement your own prioritisation mechanism, is indeterminate, and you should never rely on it.
As to what will happen, well that depends.
For two inserts to the same table, if data integrity is dependant on what order they are executed in your database design has a horrendous flaw.
For collisions (two updates to the same record for instance). There are two implementations.
Pessimistic locking. Assume there will be a significant number of updates to teh same data, so issue a lock around it. If The lock exists fail the update (e.g. second one if first hasn't finished) with some suitable message.
Optimistic locking. Assume collisions will rarely happen. Usual way of doing this is to add a timestamp field to the record which changes every update. So when you read the data you get the timestamp, and when you write the data you only do it, if the timestamp you have matches the one that's there now, and update said timestamp as part of it. If it does not match you do the "Someone else has changed this data message".
There is a compromise position, where you try and merge two updates. (for instance you change name and I change address). You need to really think about that though, it's messy, and get very complicated very quickly, and getting it wrong run's a real risk of messing up the data.
People with far larger IQs than mine spend a lot of time on this stuff, personally I like to keep it like me, simple...