Your fern is a Boston Fern and it looks like it needs more water.
- Keep the soil damp at all times
- Provide bright light (but no sun)
- Don't let it sit under or over A/C vents (they hate dry air)
- Put the pot on a humidity tray (a saucer full of pebbles or marbles that you can keep full of water - but don't let the fern sit in the water, but rather on top of the marbles/pebbles).
You can cut the dead leaves off. With proper care, the fern will completely fill the pot in less than a year.