Bitcoin Forum

I have been running Bitcoin-Qt without a problem, and some online wallets. 
Bitcoin-Qt took a few hours to get synced up. 

Do all Wallets need to download the blockchain? 

How can an Android wallet download the full blockchain when it only has a limited amount of memory?

The wallet contains a private and public key pair.  The block chain contains all the transactions which users have encrypted BTC to one another.  An application like Bitcoin-Qt, Multibit or an online wallet will use your private key to determine if you have any BTC on the block chain. 

So a "Wallet" application may or may not download the block chain to your computer, but it does need access to one to determine if you have any BTC on it.

Bitcoin-Qt & Multibit - Download local copies of Block chain
online wallets typically store the block chain on the server, so it doesn't down load to your PC.

No. Some wallets (like Electrum) use an external blockchain (IIRC, Electrum hosts their own). Your client simply connects to their servers to verify things and get updates. This is good for a lot of people, because the blockchain is huge and constantly getting bigger. However it's somewhat less secure, you have to trust an external party. By running a "full node" (downloading the whole blockchain with bitcoin-qt), you are not only making yourself more secure, but also improving the security of the network (you have the full chain, so your client can help call out cheaters, to simplify things a bit). Whether this is worth the extra computing time and 6gb of data is your call.

(You can speed up the download some by downloading most of the blockchain from a torrent and copying it into your bitcoin folder. Your client will then only have to sync the blocks that happened after the torrent upload).

No necessarily all Wallets need to download the blockchain.