First: your title says Windows 7 and in your post you mention windows 8 and windows XP. That's three different versions of Windows. On top of that, there's also Windows 8 and Windows 8.1, but I assume that you use Windows 8.1 (and not 8).
Couple of things:
- All three versions allow you to run almost every program as administrator by right-clicking the link.
- If you double-click something ("the normal way"), you run it as the user you're currently logged in with. This is good enough for almost everything.
- All three versions allow you to do the Start -> Run thing, though Windows 7 and 8 don't call it "Start" any more. The easiest way to access that Run thingy is to press the Windows key on your keyboard together with the 'r' key (r as in "run").
Running as administrator should only be necessary in "special cases", e.g. running a command prompt or some system configuration program in administrator mode to be able to issue some potentially dangerous system commands, or a sysadmin installing an application on a computer where a limited user is logged on (often the case in corporate networks).
So, the short answer is: just double click. If that doesn't work correctly and you don't know why, but it does work when you run it as administrator, you're probably doing something you shouldn't be doing, or your user accounts are misconfigured. In such cases, please ask a more specific/detailed question here and you have a much better chance to get some useful case-specific information.
Good luck!