Command+C (copy) with the left command key does not work
Make sure when you press CMD+C your Edit menu in the top flashes, this indicates it works as expected.

If it doesn't, go to Keyboard settings in System Preferences and:


Restore defaults Modifier Keys...,
Restore defaults in Shortcuts tab,
In App Shortcuts click the + button and test if Keyboard Shortcut input box will react on your combination.


If Edit menu flashes and it still doesn't work, then the first thing to do is to restart pasteboard service by the following commands:

launchctl stop com.apple.pboard
launchctl start com.apple.pboard


If still doesn't work, here are some other suggestions to troubleshoot the problem:


Re-test in different applications just in case.
Re-test if works when using contextual menu (without using the shortcut).
Re-test alternative combination such as CMD+X for Cut instead. 
Run Console to see your System log queries if something obvious is happening.
Test if the Clipboard works from the terminal via:

$ echo test | pbcopy
$ pbpaste


If Paste menu item is no more greyed out, then maybe it's a problem with Paste command.

If you're advanced user, use dtruss or fs_usage to debug it, for instance:

$ sudo fs_usage | grep -C4 MyAppWhereCopyDoesntWork


and then notice other apps which appearing exactly when you press the shortcut which potentially you could consider quitting for testing purposes.
Disable any recent programs that you run or restart your computer.
Disable VoiceOver as suggested here.
Force quit the Finder (e.g.: killall -HUP Finder).
If still doesn't work, consider rebooting the system, especially in Recovery or Safe mode to re-test.


As for a workaround, you can install Karabiner app and remap your shortcuts by configuring private.xml file (see Misc tab). For more suggestion, please check the following posts:


Remap Ctrl+C, Ctrl+V, and Ctrl-X only
How can I reassign the copy/paste keyboard shortcuts?