This plant is a Dracaena fragrans ' Massangeana' or simply a corn plant. Based on your photo the plant looks like it should be repotted into a slightly larger container. They do okay if they are slightly pot bound but if it has been more than 2-3 years since it has been repotted the plant would benefit. The one branch does look like it is drooping but you should do let it be. Also, make sure that the plant is not being overwatered. That can cause root rot.
The following is a link to information on their care: