I recently downloaded Morrowind Code patch but everytime I try to install something new I get an error message saying.

anolyzing Morrowind.exe...
Patching code...
Writing patch...
There was an error while accessing a file:
Permission denied
Patch failed to apply.

Can anyone help me out? I already tried re-downloading it.

A few things come to mind...

The MCP requires the final version of the game (1.6.0.1820) This means both Tribunal and Bloodmoon must be installed.

Are you using Windows7 or Vista? You need to have full Admin accessibility to the files. A quick way to by-pass all Vista/Win7 problems is just http://www.gamesas.com/index.php?/topic/781497-vistawindows-7-problems/page__view__findpost__p__11352753 of Morrowind.

Have you altered your Morrowind.exe in any manner? (like the 4GB patch, or Exe-Optimizer) These need to be installed after the Code Patch.

I do have the final version of the game and I'm not sure if MGE is a exe-optimizer, but I am on a windows 7 but I had Morrowind before and it use to work fine all the time.

Did you try running it as the administrator?

MGE doesn't touch the Morrowind.exe file, so it's something else that's causing your problems. Where is Morrowind installed? If it's the Program Files (x86) folder (you'll only have this if you're running the 64-bit version of Windows 7), then move it like Pluto recommended and run the code patch as administrator (right click the file and select Run as administrator).