This is probably something I don't understand since I am used to Windows 
and am only starting out with Ubuntu. I know that software in linux comes in packages what I don't understand is why can't non-admin users install software.

I mean, every application is run by a specific user, and that user will
only be able to run that applciation with his privilages, so if he has
no admin privileges, the application also won't be able to access unauthorized
directories etc.

I want most of the time to work on my PC with a non-admin user since it seems
more safe to me, most of the time I have no need for admin privileges.
and even though I know viruses in linux are uncommon I still think the best
practice is to work on the computer in a state that you yourself can't
make any changes to important files, that way viruses also can't harm any important
files, but I need to install software for programming and web-design etc.
and first of all I don't want to switch users all the time.
But also it sounds safer to me that everything being done on the PC
will be done through the non-admin user.

I'll be glad to know what misunderstanding I have here, cause
something here doesn't sound right.
['https://askubuntu.com/questions/48243', 'https://askubuntu.com', 'https://askubuntu.com/users/19733/']
Non-admin users cannot install software because the packages run as root when they're installing as they install to privileged parts of the system, run maintainer scripts, etc. 

There is currently no way to tell the system "Install firefox from this .deb but in a user's home directory so that it's isolated from the rest of the system"; which is why currently it's mostly an all or nothing affair. (This is also why running third party .debs is bad, the package and it's included scripts have root access to your system)
You touched upon a BIG difference between windows and ubuntu. In Windows when you are logged in as an admin programms will be installed without asking for a password. This enables also malware to run their programs. In Ubuntu (Linux) even being logged in as an admin the system will always ask for your password when you change the system. Therefore, malware cant intrude your system easily. To sum up run Ubuntu as an admin. If you open another user account for your kids than give them only normal user rights so they cant screw up the system.