I'm fairly certain that this is a Norfolk Island Pine tree. I haven't heard of anyone successfully propagating a new plant from a cutting, but I would never say never. Give it a go, but still visit the garden center. They are very slow growing trees and could take over 10 years to reach that size.

Bathgate's got it right. A bigger pot, more light, and some plant food would help it to grow new branches, but it may never regain the Christmas tree look it had originally.

The Norfolk Pine is an indoor plant that loves warmth, humidity and bright sun. This plant along with the Asparagus fern suspended from a macrame pot hanger were the “ fads” of the disco 1970’s along with leisure suits ( yes I had one in the closet and those plants in my apartment) and creme de menthe ( had a bottle of that too). I got caught up in the kitsch of that wonderful time.