You can install a huge number of iOS and Android apps that simulate calls to your phone. Most of these will ring your phone, spoofing a fake caller ID ("Look, it's the president again!") and then coach you through a conversation. The software tells you what to say so that others in the room will think you're responding. Some of the more popular apps include Fake-a-Call for iOS, Fake Conversation for iOS and Fake Call Me for Android.
Similar apps will send you fake text messages. For that service, try Fake-a-Message.
If you want to pretend to be someone you're not while making a real phone call, there's an app for that. Fake Voice will mask your voice, making it sound like someone else's.
A Windows application also called Fake Voice does something similar.
There's also an interesting subcategory of smartphone apps that spoof your GPS location, making it look like you're somewhere you're not. These include Fake GPS Location on Android.
If you're an Android user and want people to believe you're an Apple user, you can use the Fake iPad 3 Screen app. (Make sure your Android tablet is rectangular with rounded corners.)
Some software lets you pump pre-recorded video into webcam software to make it seem like you're someone or somewhere you're really not. PerfectFakeWebcam and Fake Webcam are two of the more popular options.
A download called Pwn Mail spoofs the sender address for outgoing email. That means you can send email, but the recipient thinks it came from whomever you specified.
Some lying software is just plain goofy. For example, an iOS app called Fake a Fish lets you take one of your actual photos and add an image of a big fish, so it looks like you caught it.