Introduction
Funds are your responsibility, so take wallet security seriously.
Security is a far-reaching and detailed topic in its own right, for which there are many good resources.
In terms of Bitcoin wallets, several will offer a range of security features, but it can be hard to know which of these offer protections that are valid for scenarios you can envisage finding yourself in.
It's important to understand one of the fundamental differences between currencies such as Bitcoin and the Dollars, Euro, Yen or Paeso that you may be familiar with: A traditional bank may be able to refund you should you lose funds during a fraudulent scenario such as an unauthorised person discovering your internet banking password or stealing or skimming your credit card. But with digital currencies like Bitcoin, the burden of security is on you. If you lose access to your Bitcoin wallet or the private keys that it securely maintains on your behalf, you have lost that Bitcoin forever. You only need to Google "Lost Bitcoin" to hear of some amazing and saddening stories of thousands of US Dollars worth of Bitcoin having been lost forever, due to poor security.
Depending on the frequency with which you use Bitcoin for payments, and/or the amount you hold or wish to hold in Bitcoin, you may choose to be more or less security conscious and therefore select from a subset of the total range of security features offered by currently available wallets.
Fake Wallet
This security feature allows you to create what appears to be a genuine wallet with a genuine wallet balance, but is in fact only a facade, containing perhaps only a fraction of your total Bitcoin balance. This confers a "plausible deniability" ability onto users in a "5 dollar wrench" attack scenario.
Pin On Wallet
This security feature allows users to set a secret PIN, which needs to be manually entered into a wallet, before being able to use it or any of its features.
Pin On TX
This security feature allows users to set a secret PIN, which needs to be manually entered into a wallet before being able to send any transactions.
Encrypted Password
A wallet with an encrypted password feature will simply store the password in question in an encrypted form on the device itself, therefore rendering a stolen device such as a smartphone, useless to a thief or attacker who isn't also in possession of the cryptographic keys used to perform the encryption itself.
Dust Protection
Unlike most decimal currencies that are divisible only to two decimal places, and usually employing a separate currency denomination e.g. Dollars and Cents, or Pounds and Pence; Bitcoin is divisible to eight decimal places, each unit of which is known as a Satoshi. This immediately allows for very fine-grained payments called "micropayments" and Dust is the word generally used to describe extremely small amounts of Bitcoin, usually in the context of receiving change from a standard Bitcoin transaction, around the 1000 Satoshi mark (0.00000001 Bitcoin).
Scrambled Pin
Wallets offering a scrambled PIN feature will scramble the digits on any PIN-entry screen, to confound screen-reading and key logging malware that could be present on a mobile device. Whenever the PIN-entry screen is displayed, the keypad will arrange each digit differently each time.
2 Factor Authentication
A wallet that features two factor authentication (2FA) will require you to use at least 2 different means of verifying that you have permission to either open a wallet or create and send a transaction on that wallet - it depends on the wallet itself as to how exactly it has implemented this feature.
Keyless
Ordinarily, almost every wallet you'll come across will prompt you to store a set of 12 or 24 words known as a "seed" or "seed phrase" or "wallet seed" the first time you use it. The seed is used by your wallet to derive a private cryptographic key, which in turn is used to generate unlimited, payment addresses - good for privacy (to prevent simple analysis of payments to the same address using public block explorers). If you lose your wallet or it's stolen, a new wallet can be purchased or downloaded and the funds associated with the former can be re-instated within the latter.
In and of itself, this is a pretty useful feature, except that you need to keep the seed somewhere secure away from the wallet itself. Often this means writing it down on a piece of paper, storing it on some sort of indestructible device or material and storing that somewhere safe like a lock box or bank vault.
With a Keyless wallet however, there is no single private key stored in a single location. Instead the key is derived from multiple, geographically separate and cryptographically produced pieces and assembled from each piece whenever they're needed, e.g. when your wallet digitally "signs" transactions on your behalf when making a payment.
Touch ID
A wallet with a Touch ID security feature, allows it to authenticate you either for a making a transaction or for opening the wallet itself, with the use of a finger or thumbprint. If you have a recent Apple or Android smartphone, you'll probably already be familiar with this feature.
Further Reading