Why won't my computer open programs?

You definitely have a Malware infection. Not being able to start the task manager is one of the classic tactics. You can't tell what is running and you can't terminate processes if it does not allow you to run it.
Start it in safe mode with networking and download and install Mallwarebytes.

This is the web site: http://www.malwarebytes.org/

Run a complete scan with that in safe mode. Remove everything it finds, its one of the better programs.

After it has removed the bad stuff you still might have to fix the damage.

Sometimes these bad programs change the file associations so that when you try to run something like an exe file the system doesn't know what to do. A person who used to work for Microsoft wrote a bunch of file association fixes. Here is the web site for those. They are registry entries, the registry editor incorporates them into your registry and they fix the particular association problem.

http://www.dougknox.com/xp/file_assoc.htm

Run Malwarebytes scans on a regular basis to catch some of the stealth programs.

If your system is XP copy the following text into a blank notepad document and save it as "XP_EXE_Fix.reg" and double click on it, allow it to be added to registry and you should be good. Then do as previous poster says and do a good virus, spyware clean. majorgeeks.com has good walkthroughs.
Begin copy here:
Windows Registry Editor Version 5.00
[HKEY_CLASSES_ROOT\.exe]
@="exefile"
"Content Type"="application/x-msdownload"

[HKEY_CLASSES_ROOT\.exe\PersistentHandler]
@="{098f2470-bae0-11cd-b579-08002b30bfeb}"

[HKEY_CLASSES_ROOT\exefile]
@="Application"
"EditFlags"=hex:38,07,00,00 "TileInfo"="prop:FileDescription;Company;FileVersion" "InfoTip"="prop:FileDescription;Company;FileVersion;Create;Size"

[HKEY_CLASSES_ROOT\exefile\DefaultIcon]
@="%1"

[HKEY_CLASSES_ROOT\exefile\shell]

[HKEY_CLASSES_ROOT\exefile\shell\open]
"EditFlags"=hex:00,00,00,00

[HKEY_CLASSES_ROOT\exefile\shell\open\command]
@="\"%1\" %*"
[HKEY_CLASSES_ROOT\exefile\shell\runas]

[HKEY_CLASSES_ROOT\exefile\shell\runas\command]
@="\"%1\" %*"

[HKEY_CLASSES_ROOT\exefile\shellex]

[HKEY_CLASSES_ROOT\exefile\shellex\DropHandler]
@="{86C86720-42A0-1069-A2E8-08002B30309D}"

[HKEY_CLASSES_ROOT\exefile\shellex\PropertySheetHandlers]

[HKEY_CLASSES_ROOT\exefile\shellex\PropertySheetHandlers\PEAnalyser] @="{09A63660-16F9-11d0-B1DF-004F56001CA7}"

[HKEY_CLASSES_ROOT\exefile\shellex\PropertySheetHandlers\PifProps] @="{86F19A00-42A0-1069-A2E9-08002B30309D}"

[HKEY_CLASSES_ROOT\exefile\shellex\PropertySheetHandlers\ShimLayer Property Page]
@="{513D916F-2A8E-4F51-AEAB-0CBC76FB1AF8}"

Open it in Safe Mode and run a complete virus scan. If nothing is found, open Device Manager and see if there are any conflcts, if there are none, open Processes and shut down any programs you don't need.
If none of those work, try to restore your system to a point before you started having problems.

 
If you can, click on the START button, type c:\windows\system32\restore\rstrui.exe
into it and press ENTER.  This will start the system restore wizard.
Select a restore point before the problem started happening and once restored, you should be able to boot.
Then run a full malware and AV check on your machine. 

Impossible to answer without someone getting their paws on your computer, I'm afraid. Could be software, could be malware, could be hardware.