This is also growing on the side of my house. I think it's pretty, but I have NO idea how to search for what it is. If it's something worth keeping, I need to transplant it. (the plant is woody, about 3 feet tall at this point. The one in the foreground - not the trumpet vine)

Maybe a Ligustrum,
http://garden.org/plants/searc...

Ligustrum vulgare?
https://www.google.be/search?q...

or perhaps crepe myrtle?

It does look a lot like Ligustrum, here's another for comparison: Chinese Privet (Ligustrum sinense)

Ditto on invasive privet.
Don't move... Dig out and toss on burn pile.

There's a strong branch shooting vertically at the back which looks like a young stem of Lagerstroemia indica with alternate leaves on the following link ..

http://www.arbolesyarbustos.co...

The other leaves don't quite look the same (maybe the angle?), although this plant can have opposite leaves too which I think the vertical branch has.

http://oregonstate.edu/dept/ld...

@Shell
Can you get a close up shot of the leaves both top & underside please? Also the stems? That would go a long way in helping to ID it.

Thank you for your replies, everyone! As soon as the weather clears up, I will take more pics.