It looks like a split-leaf philodendron.  It is fairly easy to grow, survives in medium light and normal watering. In low light settings sometimes the leaves don't split.  It gets quite big very fast - yours could use repotting. It can be propagated either by cuttings or by air layering. I would not try a root division. It grows very tall and has aerial roots that will attach to a moss pole, if you provide it with one.

Here is a general reference on houseplant care, with some specifics for a split-leaf philodendron:
http://extension.missouri.edu/p/G6510

More specific care instructions are here:
http://www.healthyhouseplants.com/index.php/plant-of-the-month/split-leaf-philodendron.html