Copying files is very simple. Just drag and drop a file on your AirDroid desktop client and it will be sent to all connected devices. I used to send emails to myself to do this.
If you don't want to install the desktop client or if you are using a public computer you can access your phone via the web interface web.airdroid.com. The web interface is even more powerful than the desktop client. You can install and delete apps on your device, change ringtones, take screenshots, manage your address book and much more. You can even take photos with your smartphone's front or rear camera with real time preview even if your phone is in standby mode!
To be able to use AirDroid you have to create an account first. Go on airdroid.com to register. You can download the Desktop client for Windows and Mac OSX there too. The Android app is available in Google Play.
AirDroid is for free, but file transfer is limited to 200 MB per month (I think). You can get an unlimited account with some extra features for 20 USD per month.