My biggest question, is, if the fork is implemented, how will 1 side of the fork be differentiated from the other side?
They will have different blocks in the blockchain and therefore different records of which transactions are confirmed.
Will all previously mined bitcoins become obsolete so to speak?
No. All bitcoins mined (or received in confirmed transactions) prior to the fork will exist in both systems. Any bitcoins mined after the fork will exist only in the system where it is mined. Transactions that are received after the fork may exist in both system or may be isolated to the system where it is sent (it will depend on the specific technical details of each transaction).
Will it require a new bitcoin client
That depends on which fork you want to participate in. If you want to participate in the fork with the larger blocks, then you'll need to upgrade your bitcoin client.
and bitcoin blockchain?
The two chains will be identical up to the point where they fork. After that each system will see it's own blocks added to its own copy of that chain and will ignore the blocks from the other system.
What is the limitation of the current block size that would require a bitcoin fork,
The current limitation it 1 megabyte per block. Given the current average size of a transaction, this work out to a system that can confirm about 4200 transactions per block. Since blocks are solved on average every 10 minutes, this works out to a system that can confirm about 7 transactions per second.
and will this be the last time it is changed?
Impossible to predict. I'd guess probably not.
Will this fork increase the total number of bitcoins to be made?
No.