WiFi File Explorer
Connecting your Android phone to your PC via a USB cable can often feel clunky. I certainly do not consider it an ideal way to exchange files between a computer and a phone. Furthermore, I've heard from people who have tried to connect a phone to a computer, but no icon showed up on the desktop. For anyone that has had such issues with the USB connection for their phone, I wholeheartedly recommend going the wireless route with WiFi File Explorer. Even if your USB connection is working, I think you'll want to start using this app instead.
To use WiFi File Explorer, you'll need to turn on your phone's Wi-Fi and then find a network you can access. Once you have successfully done that, you can go into the app. After a few seconds, it will give you a URL that you enter into your computer's Web browser. This will take you to a Web page where you can look at the contents of your SD card. It will also helpfully display how much memory is available on your card, as well as the size of the files that you are looking at.
So how does this work? Let's say you wanted to transfer some pictures that you had taken with your phone. You would go to the Web page that WiFi File Explorer directs you to, and go to the dcim directory, and then to the camera subdirectory. From there you should see all the JPEG files you have taken with your phone. Then it is simply a matter of clicking on the file in question, and saving it to your computer's hard drive. You can do this with any file on your phone, not just pictures.
For anyone worried about security issues, WiFi File Explorer also allows you to set up a password so that only you can access the Web page where your phone is sharing its SD card contents.
With the free version, you can only transfer files to your computer, which is probably enough for many people. However, if you'd also like to upload, delete, copy, and create zip files on your phone, you can upgrade to the Pro version of WiFi File Explorer for a little over a dollar.