Welcomt to the fourm!
The best way we can ID your banana is to post a picture, and even beter to id them is to post a picture of the bloom.
cutting off suckers, something about replanting using the sword suckers, full sun, partial sun, blah blah blah. Right now it's not doing anything, just sitting outside.
Suckers: are banana "pups". Suckers and pups are the same thing, they are off shoots of the mother plant that can grow into another plant.
Sword suckers: These are pups that have thin leaves when they shoot up above the ground. Sword pups tend to grow faster than a water sucker.
The full sun, part sun thing depends on the type of banana that you have. Like different plants some do petter in part sun others do better in full sun. For now just keep on doing what you are doing, keep watered (moist), and cut off thoes dead leaves. In the summer you should try planting your banana in the ground, and watch how big it will get.
Also you might want to bring in your banana before you get a frost and put it in a big west-south facing window.
Again welcome to bananas.org!! 
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A banana plant in the autumn gale -
I listen to the dripping of rain
Into a basin at night.
~ Basho Matsuo (1644- 1694)
