Modern Operating Systems operate on a permission system to try to protect your computer from harmful software as well as user errors. On both Macintosh and Windows systems there are administrator accounts that have the permissions to install software in the appropriate locations. At times a computer will complain about a lack of permission even for administrator accounts when the OS tries to do somethingn and fails. For example, when you try to download a file and there is insufficient space for the file you are downloading. The OS might mistake a lack of space for not having the permission to write the file being downloaded.
If you need help in downloading and installing Adobe Reader, please give us more information on your computer (OS, RAM, free hard drive space, and if you are trying to download and install with an admin account) together with the version of Reader you are trying to install and where you downloaded the installer file [please give the url]). This will give a better chance of helping you install Adobe Reader.