I have an IPhone 4 that my wife and I downloaded music to from my computer, when we connected and synced our computer and phone it downloaded a music album to my phone . I would like to delete this album but I'm having no luck with accomplishing this

You have to un-check the album on computer iTunes with your iPhone selected and connected then sync.

What is the name of the album? Is it possible that this is the U2 album that was provided to all iTunes owners? If you look at the album/songs on the iPhone do they appear with a cloud symbol next to them? If so, then it is no longer on your phone, just in your list of purchases.

Remove iTunes gift album "Songs of Innocence" from your iTunes music library and purchases - Apple Support

Note: The album will be completely removed, and will need to be re-purchased, if you or your wife want it back.