Bitcoin Improvement Proposals or BIPs are design features, inputs, ideas, information, or changes for essentially how Bitcoin works. Topics can vary to virtually anything regarding Bitcoin.
They are proposed to the Bitcoin community (users, miners, developers, businesses, investors, etc.) in a concise but detailed technical document.
Think of them like rules of a game. Let's use playing a game of Tic Tac Toe with you and your 4 friends (5 total) as an example.
There is Rule #1 explaining how to create the 9 x 9 grid.
There is Rule #2 stating that you must use X's and O's.
The Bitcoin community votes and the proposal is either approved or denied. If it is approved, it becomes part of Bitcoin's standard documentation.
Back to Tic Tac Toe. Down the line, you decide that X's and O's reminds you of hugs and kisses and the lack of love you received as a child.
You propose to use D's and P's and all of your friends agree because they feel your pain.
Thus, a new rule (Rule #3) is born and Rule #2 is obsolete.
BIPs are NOT always mutually exclusive. Their topics can be unrelated and more than one can apply.
In our Tic Tac Toe example, because you follow Rule #1 does not mean you don't follow Rule #2. You create a 9 x 9 grid AND use X's and O's.
Not necessarily everyone will follow all BIPs.
A group in the Bitcoin community can choose not to follow a BIP thus creating a separate group of users following the old set of rules and a group following the new set of rules.
These two sets of BIPs may not be compatible and can be mutually exclusive. One BIP can make another BIP obsolete making it impossible for the groups to follow both sets of BIPs.
The groups becomes a separate entities in a sense. This is called a hard fork.
Let's say 3/5 people in your Tic Tac Toe group do not agree on Rule #3, using D's and P's.
But your heart cannot bear to play using X's and O's.
So you and your supportive friend (2/5) decide to go off and play your own game of Tic Tac Toe using Rule #3 of D's and P's.
Your new group no longer recognizes X's and O's.
The three others continue their game with the old rules (with X's and O's).
The old group does not recognize D's and P's.
Thus, your new group and the old group can no longer play with each other.
But, it is possible for two sets of BIPs to be compatible. A new BIP may introduce a new functionality that is not necessarily required for Bitcoin's core function.
A minority group may decide not to follow the new BIP temporarily but both groups can still function with each other.
This is called a soft fork.
Let's go back and say your Tic Tac Toe group agreed and do follow Rules #1, #2 (now obsolete), and #3.
Someone in your group proposes that you all wear fedoras while you play. You are the only one to disagree because you cannot afford to purchase fedora until next week.
But the majority agreed and created Rule #4: Four-doras.
Your 4 friends begin wearing fedoras while playing Tic Tac Toe. And then there's you, fedora-less. You can still play with them. You just don't look as cool.