In a desktop/mobile wallet, the public and private key pairs are stored on your desktop/mobile device. The benefit here is that you have complete control over your wallet. However, if you lose or erase your devices, you cannot retrieve your wallet.
These wallets will allow you to check your balance, see past transactions, and help you with sending and receiving money. They often allow you to take a backup of your wallet. If you have a backup, you can restore your wallet, even when your device is erased or lost. Though these wallets are safer than online wallets, they are prone to virus attacks. Toast wallet is an example of a desktop/mobile wallet.