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There are at least two possible reasons for this:

1. You are not in the admin group
2. There is a permissions problem on the Applications folder

If you are not an admin, you need to become one (or change permissions on the Applications folder which I don't recommend).

If you are an admin, repair permissions as this should clear up any abnormality.

If that doesn't help there may be an ACL that is stopping you. This could have been put in place by your sysadmin (if you have one). We can cross this bridge when we come to it.

More details about your situation are needed. Are you the only user on the machine, etc.
 
Do you mean install an app package, or just dragging 'n' dropping an app into the folder? If you're running an installer, I think you're out of luck, I believe any installer installing an app will require the password. If you're copying an app in manually, calderone's advice above is spot on.
 
There are two admins on this computer, and I am one of them. When prompted, I am required to type in my own name and password. How do I fix it from here?
 
There are two admins on this computer, and I am one of them. When prompted, I am required to type in my own name and password. How do I fix it from here?

Open up a terminal and type in

Code:
ls -l / | grep Applications

and hit Enter. Copy and paste what comes up into a post.
 
drwxr-xr-x+ 82 kellypeters 501 2788 11 Nov 17:53 Applications

Try repairing permissions. If that doesn't fix it, these two commands should do the trick. (I think)

Code:
sudo chmod -R +0020 /Applications
sudo chown -R root:admin /Applications

Then things things should be back in order.
 
Repair permissions before toying around in Terminal.

The chown command is correct to get the ownership back how it should be. The chmod command is supposed to get admin's back write permission, but it may not be correct. I'm just guessing what it would be from the man page. I really don't use chmod often enough to know.

And what calderone said. Don't try with those scripts unless the Repair Permissions option in Disk Utility can't fix it.
 
The chown command is correct to get the ownership back how it should be. The chmod command is supposed to get admin's back write permission, but it may not be correct. I'm just guessing what it would be from the man page. I really don't use chmod often enough to know.

And what calderone said. Don't try with those scripts unless the Repair Permissions option in Disk Utility can't fix it.

Here is what I would run if it came down to that:

Code:
chown root:admin /Applications
chmod ug=rwx /Applications

I wouldn't run it recursively.
 
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