How to encrypt data on an external hard drive?

Have you looked into TrueCrypt? From the website:

Main Features:

• Creates a virtual encrypted disk within a file and mounts it as a real disk.

• Encrypts an entire partition or storage device such as USB flash drive or hard drive.

• Encrypts a partition or drive where Windows is installed (pre-boot authentication).

• Encryption is automatic, real-time (on-the-fly) and transparent.

• Parallelization and pipelining allow data to be read and written as fast as if the drive was not encrypted.

• Provides plausible deniability, in case an adversary forces you to reveal the password:

• Hidden volume (steganography) and hidden operating system.

• Encryption algorithms: AES-256, Serpent, and Twofish. Mode of operation: XTS.

TrueCrypt should do the job.

http://www.howtogeek.com/howto/6169/use-truecrypt-to-secure-your-data/

TrueCrypt doesn't work if you're planing to use backup programs such as Windows Vista/7 backup, as it does not see the drive mounted. I tried both options with no success. The solution is to look for backup drives that come with encrypting option.

You didn't mention, what system you are using to back up your data. Anyways, the free software Veracrypt or TrueCrypt (without diving into the discussion which is better/more trusty) could do the job.

If you trust Microsoft, BitLocker is also available - but it is closed source.

For Linux-systems, Veracrypt is being offered as a GUI and CLI implementation as well. Furthermore, you can use cryptsetup or related software to encrypt your hard drive. I personally like to use hardware with linux unified key setup (LUKS)-encryption, since it is fairly well documented, e.g. in the fedora wiki or the arch wiki.