A hardware wallet is a special type of bitcoin wallet which stores the user's private keys in a secure hardware device.
They have major advantages over standard software wallets:
private keys are often stored in a protected area of a microcontroller, and cannot be transferred out of the device in plaintext
immune to computer viruses that steal from software wallets
can be used securely and interactively, as opposed to a paper wallet which must be imported to software at some point
much of the time, the software is open source, allowing a user to validate the entire operation of the device.