Your plant appears to be a Dracaena  or ti tree.  While the woody trunk and leaf crown has a palm-like appearance they are not true palms.  This is an indoor plant in our area and can be quite easy to care for.  It is difficult to make out which species of Dracaena you have from the photo but here are the general instructions for care:

• Keep the plant at no cooler than 55 degrees in winter and preferably at 65 to 75 degrees at all times.  Leaves will droop and eventually fall if cooler.
• Near, but not in, an east or west window where light is filtered but bright
• Keep the soil thoroughly moist at all times; reduce water in winter but never let the plant dry out
• Mist leaves twice weekly
• Repot every other year
• Standard fertilizer application every two weeks in growing season

You can cut the broken trunk off if you wish, using sterilized shears, but if it is a healthy enough plant it may regrow. Make a clean, sharp cut at the top and watch for growth from nodes (small bump visible under the bark) that will appear near the cut. This can take several months to occur and it may not take off in leafy growth until the next season. But this is a good time in the growing season to give it a try and to make sure you are caring for it correctly so it has the best shot.

Good luck with your plant and let us know if we can be of further help.