Running a full node is not for everybody. There are lightweight clients that don't require the blockchain to be downloaded.
Simplified payment verification (SPV) clients, such as Multibit and Bitcoin Wallet for Android are examples.
Other lightweight clients, including Electrum and Bitcoin Spinner, use a client/server model, where your private keys are still safe locally. Being client/server, these clients do rely on an external party and thus trust is being placed with the server that the information about balances and confirmations is accurate though, so there are reasons to not store or transact significant amounts of coins through these methods.
And then there are hosted (shared) EWallets such as Coinbase, Paytunia and Instawallet as well, in addition to the EWallet accounts for most exchange accounts. Again, with these you are extending full trust to an external party -- thus consider this before storing larger amounts of coins with these methods.
Here's a review of some options:
- http://bitcoinmagazine.net/bitcoin-wallet-options/
As far as those needing or wanting the full blockchain, there are methods to speed up the initial startup. If you pull the bootstrap.dat torrent and run v0.7.1 of the Bitcoin.org client, you'll be up to speed much sooner.
Going forward, the next version of the Bitcoin.org client, (e.g.,likely v0.8.0), will include ultraprune, which makes a huge difference with regard to the performance of the initial block load time.