Your coins seem to be in your account on the public blockchain explorer, so I don't think you have to worry.
The next step is to work out what has happened in your local wallet. Presumably you are using the Ethereum Wallet. Check the block number (like 2,105,732 currently) on the top of your Ethereum Wallet screen. This should match the latest block in https://etherscan.io. If your block numbers are not matching, you may be on the non-hard-forked Classic chain. And looking at the Classic chain explorer, your 105.632 ETC transaction did occur. See https://etcchain.com/addr/0x108091b54a547e9334feaa72f59ec47272654dea.
I suspect that your Ethereum Wallet is synced to the non-hard-forked Classic chain.
Check the menu Develop -> "The DAO" Fork. You probably have "Don't Support DAO Fork" selected. Change this to "Support DAO Fork". You will have to wait until your blockchain re-syncs again and this will take a while.
You should check out How to conditionally send ethers to another account post-hard-fork to protect yourself from replay attacks as you could accidentally transfer your coins to the wrong address if you are not careful.
You will also have to have Ethereum Wallet v0.8.1 or above with geth 1.4.10 or later.
Response to questions in comments
Q When I make the switch over to the new blockchain will my wallet rebroadcast the same transaction again but on the new blockchain?
A It should not. The transaction was broadcasted by the Ethereum Wallet to the Classic network and some nodes on the Classic network may or may not relay the transaction to the hard-forked network.
Q Will this not mean that the same transaction is in two different block on both chains or can my wallet edit a block from the past?
A The same transaction can be in two different blocks on both chains. Your wallet cannot edit a block from the past.
Q I have clicked the link to view my transaction on the old blockchain however the link does not work as the website is down. Is there another website that allows me to check my transaction? I wanted to check I sent the coins to the correct address? -> 0x6b163A9A8804d2e8F76F7FdC4EaD042393631481
A I've updated the link to a work Classic block explorer.
Q Also, what chain is dominant? If I have coins in my wallet on the new chain and no coins in my wallet on the old chain what chain wins? What chain is dominant?
A From http://fork.ethstats.net/, the Classic chain has ~ 14% the hashpower of the hard-forked chain. The dominant chain does not depend on the coins in your wallet.
Q @BokkyPooBah I have one more question that has come into my head. The first transaction appeared on both blockchains both new and old. The second transaction appeared on only the old. I take it that the chains talk to each other to update each other. Is this not a glitch in the way they talk to each other as it seems odd that the first transaction would get synced and the second would not. etherscan.io/address/0x108091B54A547e9334FEAA72F59ec47272654dea
A Some nodes on the network have been configured to re-broadcast the transactions from one network to the other. From my past transactions, the re-broadcasted transactions appear on the other network with a delay of up to tens of minutes OR not at all.