Put plainly, wETH is "wrapped ETH" but let's start by introducing the players.
FIRST, THERE'S ETHER TOKEN
Ether or ETH is the native currency built on the Ethereum blockchain.
SECOND, THERE ARE ALT TOKENS
When a dApp (decentralized app) is built off of the Ethereum Blockchain it usually implements its own form of Token. Think Augur’s REP Token, or Bancor's BNT Token.
FINALLY THE ERC-20 STANDARD
ERC-20 is a standard developed after the release of ETH that defines how tokens are transferred and how to keep a consistent record of those transfers among tokens in the Ethereum Network.
WHY YOU NEED WETH
ETH DOESN’T CONFORM TO ITS OWN ERC-20 STANDARD.
As mentioned above, ETH was the proto-token of the Ethereum Alt tokens, which means it was built before the ERC-20 standard existed.
WRAPPING ETH ALLOWS YOU TO TRADE DIRECTLY WITH ALT TOKENS.
The reason you need wETH is to be able to trade ETH for other ERC-20 tokens on decentralized platforms like Radar Relay. Because decentralized platforms running on Ethereum use smart contracts to facilitate trades directly between users, every user needs to have the same standardized format for every token they trade. This ensures tokens don’t get lost in translation.
READY TO WRAP?
SO THIS IS AWKWARD...YOU DON'T ACTUALLY WRAP ANYTHING.
When you "wrap" ETH, you aren't really wrapping so much as trading via a smart contract for an equal token called wETH. If you want to get plain ETH back you need to "unwrap" it. AKA trade it back for plain ETH.
WHATEVER, JUST GIVE ME THE WETH!
Go to Radar Relay or 0x Portal to use their interface. Once you have the wETH you can freely trade with anyone holding an ERC-20 token.
THE FUTURE OF WETH
HOPEFULLY, THERE'S NO FUTURE FOR WETH.
Steps are being taken to update the ETH codebase to make it compliant with its own ERC-20 standards. Weird, we know. Additionally, ERC-20 may be replaced by other standards as problems and solutions arise. There's already a ERC223 in the works.