For most people, their greatest security interaction in their everyday life is with passwords. Most internet users have grown accustomed to handling a multitude of passwords for different services. Unfortunately, we humans are bad at remembering stuff, and this causes most of us to reuse passwords. This is bad!
In the last year, a service from Troy Hunt called HaveIBeenPwned has become much more widely known, and has shone a light on the issue of password leaks. When a service is hacked, or otherwise lose their data, your password can be among the information that is suddenly available to anyone. If you have used the same combination of username/email and password anywhere else, that account has also become compromised. While the obviously best solution to this problem is to not reuse passwords, this may not be feasible to everyone, and it still does not protect your account should your unique password be leaked.
Two-factor authentication to the rescue! As the name implies, this sets up a second "factor" that must be known or accessible in order to log in. Most used is a SMS-message to a known phone number containing a 6-digit code. Even if an attacker knows your username and password, they would still need access to your phone, greatly reducing the chance of them being able to log in.
Unfortunately, text messages are not the most secure form of communication. They are sent unencrypted on the network of the carrier, and several attacks exists to gain access to two-factor SMS codes. One involves porting the phone number in question to another carrier using social engineering. Therefore, better options should be considered.
The most accessible solution that greatly increases security is the use of Time based One Time Passwords, known as TOTP. When you sign up for a service offering this, you will be shown a QR-code that can be scanned with e.g. Googles "Authenticator"-app, Twilios "Authy", or any other app implementing the open TOTP-standard.
When you need to log in after this, you will be asked to provide a 6 digit code generated by the app. The code changes every 30 seconds. The reason this can work is because you and the service you are logging into share a secret, and you both know what time it is. Through some clever calculations, you can both compute the same token, but be sure that no one without the secret can do the same.
Better yet is the use of a physical device such as a YubiKey. It looks like a USB-stick and can either be permanently installed at the machine you usually use, or it can be kept on your keychain to be available wherever you go. When you log in to a service where the YubiKey is enabled, the physical presence of the device is checked, providing a strong likelihood that the log in attempt is legit. The barrier to entry can be quite high for YubiKey, as you would have to get accustomed to carrying it with you wherever you go, should you wish to use a certain service.
The biggest downside to using two-factor solutions is the need for storage of recovery codes. Should you lose access to the two-factor device, that being your phone or your YubiKey, you will need the recovery codes to regain access to your accounts. These must be stored in a safe location. The best option is simply to store them in a safe place in your house, or sync them to an encrypted cloud service. The worst mistake you can do is to ignore the message when you activate two-factor authentication that implores you to save your recovery codes. Without these, your accounts may be lost for ever!
All in all, two-factor authentication provides a huge improvement in security for your accounts, and you should activate this on every service that offers it. For more sensitive services like finances or systems containing a lot of personal information, you should consider using a hardware two-factor device like YubiKey for maximum protection. Just remember to store your recovery codes in a safe location!