This page is in serious need of an update. It should probably be renamed or even replaced with "Wallets". (In Edit mode don't worry about the languages/translate/T:x tags, they can be sorted by someone familiar with them later. Translation is way behind anyway.)
The Daedalus Wallet (origin) is an open-source highly secure wallet for the Ada cryptocurrency. Download and install it so you can use it to safely store your Ada. Daedalus will add more cryptocurrencies and be developed over time along with Cardano, to become a universal wallet, blockchain application platform and an app store.
Daedalus is available for Windows and Mac with basic features, with planned release for Linux and with support for other currencies such as Bitcoin and Ethereum Classic, as well as support for custom plugins and automations.
If you encounter an issue with Daedalus, head over to the FAQ page, where most issues have a solution.
Cardano is currently working with Ledger and a third-party firm in order to enable an ADA integration for Ledger's hardware wallets, such as the Ledger Nano S or the Ledger Blue. Expected to be part of the Shelley release sometime in Q2 2018.  These Ledger wallets have a display where the user can enter a pin to open the wallet and where the transaction can be verified before being signed.
A paper wallet is a new type of wallet that will be available in Daedalus as part of the Shelley (origin) Release.  Paper wallets are single address wallets (not HD wallets) that can be used for secure, long-term holding of Ada, placing funds in cold storage on a physical medium. Optionally, paper wallets can be created with a quantum resistant signature scheme that will make them secure even when quantum computers become available.
Users will be able to print their paper wallet certificates, which will contain the encrypted private key of the wallet and a mnemonic phrase that is needed to decrypt the private key during wallet restoration. The paper wallet certificate will also include an address of the wallet that can be used to receive Ada while the wallet is stored offline.
A private key can be removed from Daedalus for paper wallets stored in cold storage. After the removal of the private key, the paper wallet can still be used in Daedalus in a read-only mode. This will allow users to check their wallet balance and see the transactions while keeping the wallet safely stored offline.
Comparison of Ada wallets and exchanges
For a comparison see the comparison of Ada wallets and exchanges