I have Windows 7 installed on my PC right now with a number of programs. Will I have to install second copies of all these programs if I install Ubuntu as a secondary OS to run on the same PC?
You should first consider that your programs probably aren't going to be the same as the ones you installed in Windows. As between most different operating systems, there is little native compatibility so if you're currently using something that doesn't have a Linux or Ubuntu version (more and more applications do these days), you'll have to look for an alternative.
But assuming we're talking about an application which is available for both Linux and Windows, the short answer is yes, you'll need to install a separate copy.
When you install Ubuntu, you are installing a totally different OS with a different structure of binaries, a different kernel, a different set of rules.
Depends of the set of programs you are talking about, but there is always alternatives in the Linux/Ubuntu world to almost any Windows program that runs natively on Linux/Ubuntu. If you could not survive without them, you could use Wine (Wine is not an Emulator) which provides a compatibility layer so Windows programs could run on Linux. Of course, you will need to install them into the Linux environment.