There are several possible problems you may be experiencing, from the information in your question and in comments.
Web-E's answer explains how to install Ubuntu alongside Windows. I recommend doing that--it seems to be what you intended to do first.
If you did want to install Ubuntu inside Windows with Wubi, and you are using Windows Vista or later, the
permission denied error can often be fixed by right-clicking
Wubi.exe and clicking Run as Administrator. Running
Wubi.exe with normal abilities when UAC is enabled will fail.
If that doesn't work, then you must consult the log file that the error message referred to, to see what is wrong. It should have been created in the same folder that contains
Wubi.exe itself. If you can make sense of what the log file tells you, you may be able to solve the problem. Otherwise, you should edit your question to include the text from the log file, or post it at http://paste.ubuntu.com and add a link to it in your question. (You can then comment here to notify me.)
With all that said, remember, unless you want a Wubi system, I recommend you install a normal Ubuntu system alongside Windows by creating and booting from a USB flash drive (or CD/DVD) as explain in Web-E's answer.
Inability to Boot Ubuntu After Installing Alongside Windows
Web-E and I had both thought you meant that when you created the bootable USB flash drive to install Ubuntu alongside Windows, you were unable to boot from it and start the installation.
However, if the problem is that you can install Ubuntu but when you try to boot it, it just boots into Windows, this is caused by Ubuntu's boot loader, GRUB2, not being installed to the master boot record. To solve the problem:
Or if you prefer, you can instead any of these methods: