A Hardware Wallet Is Special Type Of Bitcoin

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
This page is an attempt to summarize all the known developments of hardware wallets that can use Bitcoin as part of their operation.
Security risks
To date there have been no verifiable incidents of Bitcoins stolen from hardware wallets. Hardware wallets are relatively new, but at least for the time being they have maintained a good track record, unlike the numerous incidents of Bitcoin theft from Internet-connected computers.

However, it's important to understand that hardware wallets are a high value target and depend on various assumptions holding true to maintain security. They are not a silver bullet, and there are several realistic ways in which a hardware wallet can fail to protect your Bitcoin. These risks need to be carefully considered when deciding how much trust to place in a hardware wallet, and which hardware wallet to buy.

How a hardware wallet could fail to protect your Bitcoin:

Malware swaps recipient Bitcoin addresses: a hardware wallet won't protect you from being tricked into sending Bitcoin to the wrong address. For example, malware on a PC could monitor for high value transactions and then swap out the recipient's authentic Bitcoin address for an address controlled by the attacker. When the stakes are high, multi factor (e.g., over the phone) confirmation of a recipient's Bitcoin address is recommended.
Insecure RNG (Random Number Generator): hardware wallets rely on the security of an RNG, often embedded in hardware, to generate your wallet's private keys securely. Unfortunately, it is notoriously difficult to verify the true randomness of the RNG. An insecure RNG may create wallet keys that can later be recreated by an attacker, by generating psuedo-randomness that would seem statistically indistinguishable from true randomness yet still be predictable to an advanced attacker. An RNG may become insecure as a result of malicious weakening or an unintentional mistake. This failure mode is common to any wallet generation procedure in which the true randomness of the source of entropy being used can not be verified.
Imperfect implementation: the security of all computing devices relies on the quality of their implementation. Hardware wallets are no exception. Bugs at the software, firmware or hardware level may allow attackers to break into a hardware wallet and gain unauthorized access to secrets. Even if the design is perfect, proving the security of a hardware or software implementation is a very hard, mostly unsolved problem. To date, no wallet in existence is implemented using provably correct software.
Compromised production process: even a perfect software and hardware implementation of a hardware wallet would be vulnerable to a corrupt production process that introduces intentional or unintentional holes into the final product. The introduction of hardware backdoors is a real concern for high risk financial and military applications.
Compromised shipping process: a compromised fulfillment process may substitute or modify secure devices for superficially identical but insecure replacements. Government programs that intercept hardware and modify them in route to insert backdoors are known to exist.
In summary:

While not a silver bullet hardware wallets can still be extremely useful, assuming you take care to use a good one: an authentic device manufactured by trustworthy, technically competent security experts with a good reputation (e.g., TREZOR).
Cold storage solutions implemented with open source software and general purpose hardware (e.g., BitKey, Pi Wallet), using a verifiable source of entropy such as physical dice may provide superior security for some use cases (e.g., long term savings).