Online, mobile or a hardware wallet, all types are available for Ethereum, for storing and processing Ether. Quite a few of them already support Ethereum’s specialty - smart contracts. There is also an official Ethereum wallet.
Mist - the Official Ethereum Wallet (Desktop)
Mist is a desktop wallet that stores private keys on your computer. After it synchronizes with the network, you are required to set a password to start using it. The password is your only access to the insides of your wallet, so don’t lose it.
Mist runs on Windows, Mac and Linux and no registration or personal information is required to use the wallet.
Besides sending, receiving and storing ETH, the official Ethereum Wallet is the gateway to daps on the blockchain, as it allows to create, deploy and use smart contracts.
Trezor (Hardware Wallet)
Trezor was first created to be used with Bitcoin, but it can be used for Ethereum as well, with the use of MyEtherWallet web interface.
Ledger Nano S (Hardware Wallet)
Another hardware wallet where Ether is stored offline directly on the device. Whenever you want to spend Ether, Ledger Nano S signs it using the private key stored on the device.
Jaxx (Mobile Wallet)
The multi-asset wallet by the Canadian company Decentral, supports multiple cryptocurrencies and ETH is one of them. Jaxx is elegantly designed and features robust security. Private keys never leave the device. Jaxx supports smart contracts.
Jaxx wallet runs on desktops on Windows, Mac OS, Linux or as an extension in Chrome and Firefox, it also has a mobile app for both Apple and Android.
Cryptocurrency wallets, like the ones below, help you store your Ethereum coins so they're not exposed on an exchange.