Can a standard user install applications?

I've been told that when a standard user installs applications, he must provide a admin user name and password, which presesumably he wouldn't have. If the admin user is standing near and does provide that info, then great, a standard user is able to then install the application, right?

My real question: Would that application be available to all of the users on that mac or just to the standard user? My research has lead me to contradictory answers and a fundamental, lingering question: If the admin account's user name and password were provided when logged into the standard account, shouldn't that application behave the same as an app that was installed when logged into the admin acccount? And, if that's true, isn't the purpose of a standard account somewhat less valuable than we think? Any insight would be of much help, thanks!