How to Recover Deleted Files

We've all accidentally sent a file to the Recycle Bin (or Trash on the Mac). Thankfully, operating systems have known for years that when you drag a file to a garbage can icon, it should not instantly delete that file. You can generally open that trash icon, find your file, and restore it to its proper place on your hard drive.
The problem comes after you've emptied that recycle bin. Then you're in trouble. But all is not lost.
The first thing you should do is check your backups or sync services. That's the best way to get a file back, fully intact. On Dropbox, for instance, there are multiple versions of each file—you can even get back a version of a file from days before you lost it.  Go into the Web interface at Dropbox.com, click the trash icon at the top with the tooltip that says "Show Deleted Files." They'll appear as part of the list of files you know and love. Right click it to restore the file, or view the previous versions of it
Those with Windows 8 and above have a tool called File History, which can help. However, for some reason, it's not enabled by default. If you've got Windows 8 and you're prone to this kind of file loss, turn this on right now by going into Control Panel, searching File History, and clicking Configure File History Settings. You need an external drive of some kind, even just a small USB flash drive, to effectively use this backup. (But remember, this isn't going to help you if you turn it on after the file has gone missing.)
With Windows 7, if you've got System Restore enabled to create the occasional restore point, you may be in luck. Windows 7 system restore creates "shadow copies" of files, so if you know the folder and the file's name, it could still be there. But you can't get to them directly. You need to right click on a folder and select Properties, then you get a tab at top called Previous Versions.
Mac users have something similar called Time Machine—it's a built-in backup that keeps files on an external drive (connected to the Mac or your AirPort Extreme router), or a network attached storage (NAS) device that supports what Apple calls Time Capsule. Either way, you've got to set up these services for them to help you at the point of accidental file deletion. They won't help after the fact.
What about getting back a file if none of the above can help? If you have a solid-state drive, we have bad news. They delete files immediately, without hesitation. If you don't have a backup of any kind and your main drive is an SSD, you're hosed.
But if you've got a regular hard drive, then all may not yet be lost. Maybe.
Recover Tools
The way traditional, magnetic platter hard drives work, you haven't actually lost that file until the space where it lived on the hard drive is over-written by other data. So there's a chance you can still get the file(s) back. That chance diminishes a bit if the PC has been used a lot since the file was deleted. (So stop using the PC.)

You'll need some special software. For Windows, there's plenty of it out there, such as Recuva (pictured above; free for home use, or $24.95) and Undelete Plus ($39.95). The problem is, if you install it after you lose a file, you might be over-writing that space on the hard disk drive with the very software meant to resurrect it! Instead, go to a different computer and download the portable version of Recuva and put it on a USB flash drive. That can be plugged into the computer where you want to find the deleted file. That way nothing gets written to the drive on the PC when Recuva is launched.
If you need similar tools on other operating systems, the open-source TestDisk supports Windows, MacOS, and Linux, and it's totally free. Its companion program, PhotoRec, specializes in recovering media files like video and photos, though it'll look for any kind of document. Neither of them really have a nice graphical user interface, but they're probably more powerful because of it.
If you want to go even further, shut down the computer with the missing file, go to another PC, and create a recovery CD/USB drive using a third-party operating system like Ubuntu. That CD could be used to boot the computer, then you can use Ubuntu tools like ntfsundelete to look at the hard drive and hopefully get the file.
If that fails, you need the big guns. Extract the physical hard drive and send it off to a service like DriveSavers. It has been professionally recovering data from hard drives for years—even drives that have had catastrophic failure or other problems. It's expensive, but DriveSavers and companies like Kroll Ontrack, The Data Rescue Center, and Salvage Data Recovery are the experts who will recover the file(s) if it's at all possible.