That is normal behavior for an Orchid. Each Pseudobulb (the tuberous bulb like lower portion with leaves on top) will shoot out flowers once, then go dormant. The remaining back bulbs (those that have already flowered) will remain and actually help the orchid by retaining moisture and nutrients that can then be used by the active growing portion of the plant. Once the Pseudobulb has flowered, one or more new growths will form at the base and become the new active growths. There are several opinions on how many pseudobulbs you should have before dividing the plant, but each division should have at least 3-4 with at least one actively growing.

The most frequent cause of Orchid death is over watering. You should allow the growing medium to completely dry out between waterings. I usually let it go a few days bone dry before wattering, and then completely drench the plant. Since the roots typically are exposed to the elements, they have a special coating, called velamen, that not only protects the root but absorbs moisture so the root has more time to absorb it. When dry, the roots will look white, and when wet will become translucent and look green.

Not all orchids behave the same, so the above is kind of general.. but for your plant, I would suggest you leave the new growths alone to do their thing. After they flower, each one will shoot out another pseudobulb, which will flower, etc.. At some point, when it outgrows the container or your planting medium breaks down, divide the plant into two or more and give one to your mom :)

Cheers,
Brad