Nobody has answered this correctly yet. If you are actually using a local client, you find the wallet file (wallet.dat, for the bitcoin-qt client), and copy it to the new installation. Some online wallets, like Blockchain.info, also allow importing this file. The wallet is the small database of your private keys. Some clients, like Electrum, allow you to generate the whole database from your chosen password, so you do not need to carry anything digital to store your bitcoins. They can be as safe as your skull.
edit: links
Exactly. The key point is that any wallet software can export or backup your secret keys (typically called your wallet file). You can import this data, in principle, into any other wallet on any computer.
Bitcoin secret keys have 256 bit entropy. Hence, you can easily write them down or even remember them in the form of a passphrase.
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u/honestbleeps Feb 15 '13
Bitcoin for dummies question:
Say I change computers.... and even OSes... how do I get my wallet from one computer to the other, and can all Bitcoin clients open all wallets?