This plant started getting dry, but I have no idea what is this plant called so I don't know how to treat it the best way.
It looks like a "Peace Lily", the common name for "Spathiphyllum" which is the plant's genus name. They are a member of the Arum family. I am not sure of the "dry" problem you might be having with your plant, but they are quite easy to care for. Here are a few links that might help more.
It looks like a Peace Lily. The damage to the leaves might be from different sources. Either it has too much direct sun light or the water is not enough. It looks a bit damp, if you water it, it should "revive". The brown spots won't go away though. Thr brown spots might also appear from watering with tap water.