When I try to open a program, i.e. Internet Explorer or Google, nothing happens. I cannot open either one.

When you say that you are unable to open any program shortcuts for example Internet Explorer and Google Chrome, do you mean this problem is specific to these browsers only?

This issue could be caused if the desktop shortcuts are broken or corrupt that means some files might be missing for that particular program.

To understand the issue and help you for this issue, I need some more information:

What is the make and model of your computer?
Did you make any changes on you computer prior to this issue?
Are you able to use the programs pinned on the taskbar? (Including File Explorer and browsers).
Do you receive any error message/code while opening the icon?

As a workaround, I suggest to follow the method below and check if it helps.

Method 1: Run SFC scan.
You can also refer to the following Microsoft article to repair corrupt files using SFC tool. Use the System File Checker tool to repair missing or corrupted system files
http://support.microsoft.com/kb/929833/en-us
After running SFC scan check if the issue is fixed, if not please proceed with method 2.

Method 2: Perform clean boot.
A clean boot is a troubleshooting technique that allows you to get the computer up and running so that you can perform diagnostic tests to determine which elements of the normal boot process are causing problems.

Refer: http://support.microsoft.com/kb/929135

Note: After performing the troubleshooting steps for clean boot refer the section “how to reset the computer to start normally after clean boot troubleshooting”.

After trying the above method if the issue still persists, please follow the method below and check if it helps.

Delete and add the shortcuts of the effected programs and check if it fixes the issue.

Step 1: To delete the shortcuts.
Go to the desktop screen.
Right click on the icon needs to be deleted.
Click on delete.
Once you delete the shortcut it should not reflect on the desktop screen.

Step 2: Add the shortcuts back on desktop and check.
1. Press Windows key + F and search for the required program or file.
2. To create shortcuts for the program on desktop.
3. Right click on the program name select "Send to”.
4. Click Desktop (create shortcut).