Need help with plant identification

Chris and I bought two house plants. We had been talking about bringing some more greenery into the house, so finally we went for it. We've always had a lot of viney plants around the house, and those grow like crazy, but we've never had the more tree-like plants. I'm finding that I have no idea how to care for them, which, I believe, is causing them to die. I'm wondering if they'll even survive because we keep our house fairly cool in the winter, and we don't have a lot of sun coming in, but I'm going to do my best to keep them alive! Any advice is welcome!

I think I've managed to identify the first one as a Corn Plant:



However, I have no clue what this one is:
Can anyone help?