The stock Android Beam functionality introduced in v4.0 is cool, but it hasn't always been very easy to use. Not all devices have great NFC chips to pair devices (I'm looking at you, Galaxy Nexus) and the selection of content you can share is limited in most situations. That's where SuperBeam comes in. This app improves upon the stock Beam experience in every conceivable way.
First, SuperBeam can be used to transfer any file between devices -- it can even move multiple files in a single batch operation. Just select the files in an app that supports the standard Android sharing intents (or use the built-in file browser), and choose SuperBeam. Both devices need to have the app installed, but the rest is automatic. Tap the devices together, and your files fly across.
Android Beam uses Bluetooth to transfer files, which is fine, but slow. SuperBeam has two modes that it will utilize as needed. If both devices are on the same WiFi network, SuperBeam just uses that connection to push the files. If they aren't, SuperBeam creates a WiFi Direct connection between the device without a router. All Android devices have supported WiFi Direct since Android 4.1, but none of the stock apps use it. This can send files at upwards of 30Mbps, which is great for large video files and pictures. You can also access the file transfers on a desktop computer via a web browser.
Even if one of your devices doesn't have NFC, SuperBeam can be used with QR code scanning or key sharing to negotiate the connection. This is my go-to file transfer utility and I install it on everything. The basic functionality is free, but a $2.99 paid version removes the ads and includes some more control over file save locations. This is unquestionably one of my favorite apps of 2013.