Hello,
on a new computer, Bitcoin Core 0.9.0 says that it synchronizes with the network for several hours now.
It also says that we are 100-something weeks behind. (Behind what?)
Behind the most recent block in the blockchain.
Does Bitcoin Core doewnloads the whole 14 GB blockchain to my hard drive?
It's bigger than 14 GB (nearly 20 GB), but yes, it downloads the entire thing and verifies every block and every transaction to make sure that they all follow the rules of the protocol.
Do I have to wait so long for synchronization to complete any time I move to a new computer?
No.
If you were running Bitcoin Core on an old computer, you could copy the APPDATA directory to the new computer. Then the new computer would already have the blockchain.
You could also use a wallet that doesn't require the entire blockchain (such as MultiBit, Electrum, or blockchain.info).