yep, draceana type family, probably one that is commonly called a "corn
plant" don't remember the proper name right this second.
looks like a humidity problem, not a watering/dying problem. that is very easy to remedy. get a tray, about an inch or so deep, about an inch or so larger around than your
plant, put some rocks in it so the
plant doesn't sit down in the tray, but rather ontop of it, then fill the tray with water, making sure the water doesn't touch the bottom of the pot. when the water evaporates, the
plant will be happy because it will ge natural humidity.
as for the old/brown/"dying" leaves, just cut them off at the base of the leaf. won't hurt a thing.