To determine what walls are weight bearing, you need to go first to your basement (or crawl space) and find the beams that will go normally from one side of the house to the opposite side (usually the shortest side-to-side dimension). These are either steel I-Beams or 4X12's or larger, supported along their course by columns (loley columns they are often called) or posts. Upon these running at 90 degrees will be your first floors floor joists, usually 2X10's or 2X12's. Some of these may be doubled or tripled together to support a wall above it, but this does not necessary mean the wall is weight bearing....only that the joists have been doubled to support the weight of the wall. The real weight bearing walls will be immediately above the basement (or crawl space) beams, and will often, in turn, support attic or second story walls or ridge beam supports. Now, if the short stub wall you speak of is built over a beam this doesn't mean you can't remove it. However, if you do, you will need to install a header that will support the ceiling joists of the first floor and will tie in with adequate weight bearing king studs in the wall.
My suggestion is that unless you feel very confident in your ability to assess this situation and take the right steps, that you consult a residential structural engineer for advice before proceeding. It'll cost a bit but may be worth it in the long run.