I downloaded some songs using an app so they weren’t stored in my music library, but into the app’s library. I want to know how I can have these songs on my computer.
I think you should have the access to the app folder first then move the songs from the app’s library to computer. iTunes may not help do this. You can try syncios, a free one, can access the library and transfer the files between your idevice and computer.
If Tim's suggestion does not help, you can try Sharepod or iBrowse to try to locate the files on your iOS device from your computer or Mac. http://www.getsharepod.com/ or http://www.ibrowseapp.com/
This should be pretty straightforward, and basically requires the latest version of iTunes, which should come with all Macs, and is downloadable on Windows from here: http://www.apple.com/au/itunes/download/
Plug in your iPhone, launch iTunes and click iPhone in the top right. If this is the first time you've used iTunes, you will need to pair the two. Once done, make sure the iPhone is still selected and click "Apps" along the top bar.
Scroll down to the field below apps where it says "File Sharing". You will see a bunch of apps in there, click the app you used to download your music and you should see the songs listed in the panel on the right.
You can then click (hold Shift+Click to select multiple) your files, and choose "Save To..." in the bottom right to save them to a location of your choice. If the app you are using doesn't appear in this list, it doesn't support file sharing and you basically won't be able to do this.
Good luck, let us know if that worked – and also which app you used!