This page is about encryption of the whole device. It is NOT about any encryption "apps" that you might download and use.
If your smartphone or tablet is not encrypted, it is very easy to gain access to the data it contains by breaking it open, and plugging it's storage "disk" into another phone, or device reader.
In the event of a targeted theft, remote wipe would not be of any use if the thief dismantled the device and took out its storage.
To protect against this, you need to encrypt the storage device itself. The most popular smartphones and tablets support whole device encryption. For Apple and Android devices, configuring this kind of encryption is very easy.
Apple iPhone or iPad
These are immediately encrypted when you set a PIN to lock the screen. That's all you have to do.
Android devices
For android devices, there is an option in the settings that enables you to encrypt the device. For more details, refer to the relevant manufacturer or network provider support pages for your device or contract.
For example, instructions for encrypting a Google Nexus device can be found here:
Encrypting a Google Nexus device
Windows phones
Windows phones have not been easy to encrypt. Contact Microsoft for help if you use a Windows phone.
Other devices: investigate "how to encrypt" before you buy
Tablets and smartphones are a changing marketplace. It is not possible to list encryption for them all here.
Whatever type of device you choose, use ease-of-encryption as one of your selection criteria.