Why you're asked for a password
Most software is designed to touch sensitive files, i.e. sensitive to the security of your private data or the systems integrity. This is why software installation is a potential risk and should be validated by a user who knows what he is doing. Even for open source Software you can not be sure that no bad code bits arrive with your percious new audio player until someone checked. And even then something could have been overlooked or packages could be manipulated on the way. Who knows what's hidden in the depth of the program? One user should.
Ubuntu provides a comfortable way of installing software. Software developers can rely on that comfort and make the assumption that synaptic/software center/apt enables them to access these sensitive files. Canonical checks the software repository for bugs and malicious code. But the ultimate responsibility is yours.
If the software in question does not need access to sensitive files, it can (in principle) be installed in the home folder though not in the Ubuntu way. You will for instance have to compile the source code yourself or run a custom install script. Beside the greater efforts of that it has the disadvantage of not allowing other users access to your just installed program (as they have no right to access your home-folder). They will need to installed it a second time. So this way of installation makes no sense in a large scale and in a small scale it's usually easier to type a password than to install software manually.
So that's why Synaptic aaks for a password and why it's good that it does.
Sudoers
If you are really in dire need of having other users install software without password, you can add them to the sudoers list. This however will result in a great security risk. If that doesn't concern you, consider that there a bot networks out there with great resources to break into your computer via Internet. They do this just to add your computer to the net and use it's connection and computing power without your knowledge to do all sorts of illegal stuff. They are not even after your personal data your you for that matter. They just want to hijack your PC. Still not concerned? Really? Then look at the following answer which is a small how-to on the workings of sudoers list:
How to make Ubuntu remember forever the password after the first time
Read that carefully. You could lock yourself out of the system.
The scare is over
So now you have the scare behind you are and taking the matter seriously, I can tell you it's really not that bad. If you trust the people working on your computer, and you don't have programs installed that allow remote access to your system (e.g. an ssh- or ftp-server), then it's really not that dangerous to disable the password stuff. Just don't do it without considering the worst case and think of your private data.
Overlook on the proceedings (don't do this lightly - see text and link above):
# in shell type these commands
sudo su # in case you do something stupid, you'll have a root shell open
select-editor # (optional) will allow you to select text editor
visudo # secure way to open sudoers list
An editor will open ADD a line like this:
confus confusion=(root) NOPASSWD:/usr/sbin/synaptic,/usr/bin/software-center
Syntax explanation: username machine=(usernameToRunCommandAs) command1,command2,.... So the above line allows confus to run synaptic and softwarecenter as root without password query. You will still need to start it with sudo synaptic or gksudo synaptic or put an alias `alias synaptic='sudo synaptic' in your shell profile.
The line you added was overridden. From man sudoers:
When multiple entries match for a user, they are applied in order. Where there are multiple matches, the last match is used (which is not necessarily the most specific match).
In your case nicholsonjf was a member of the group sudo so for him this line applied:
%sudo ALL=(ALL:ALL) ALL
If you want to override entries in /etc/sudoers just put the new entries after them.
The new entry should look like
myuser ALL=(ALL) NOPASSWD: ALL for a single user, or
%sudo ALL=(ALL) NOPASSWD: ALL for a group.
Best Answer
Sudo may be the more simple method, but PolicyKit is the more secure solution, afaik. You should go with the method outlined in the question you linked to.