Are really all transactions (the whole blockchain) stored on every client? I've read through some explanations saying that every client in the network knows every transaction because when someone wants to perform a transaction, he sends that informationen (how much, to whom and his own address) to everybody he knows and everybody they know and every.... and so on. Until it gets confirmed by everybody in the network. Then the transaction can take place. So everybody in the network knows that this transaction happens, and add the new block to their chain.

Does REALLY everybody have the whole history of transactions (a.k.a one HUGE chain of blocks) stored on their (lets say) computer?

Really all? Isn't this amount of bytes too much for normal hard disks (in my computer, smartphone, ...) considering ALL transactions which have ever happened and happen?
Everyone running a bitcoin Core node must download the entire ledger of transactions listed on the blockchain... But the vast majority of Bitcoin wallets are simply a place that stores a user's private key and allows users to interact with the network. When someone sends a transaction from their wallet, it gets broadcast to the network and eventually to Bitcoin miners. 

Once a miner finds the next block (that includes your transaction), the solution gets broadcast to the network and those that are running the full Bitcoin Core software incorporate that block into the blockchain.

So although not everyone needs to download the full bitcoin blockchain, there are still a very large number of people who do, and by doing so, secure the network.