It's an Epiphyllum hybrid. Can't tell which one - there are thousands of them. Look for "ackermannii" and "red orchid cactus", I think you will find photos of flowers that resemble those of your plant.

That's it! Thank you so much!

You have an Orchid Cactus or Epicactus I searched the web and found a good photo of a flowering one.You're in for a treat:
From the web: ARE ACTUALLY USUALLY NOT TECHNICALLY EPIPHYLLUMS AS THEY ARE ONLY USED ORIGINALLY IN ABOUT 10% OF HYBRIDIZING. THEY ARE USUALLY A HYBRID OF OTHER EPICACTUS INCLUDING, HELIOCEREUS, NOPALOXOCHIA, SCHLUMBERGERA, PYHLLOCACTUS, AND RHIPSALIDOPSIS. THEY DO NOT LIKE TO DRY OUT (BUT DO NOT EXCESSIVELY WATER), EXCEPT FOR WHEN THEY ARE DORMANT (NOVEMBER & DECEMBER). WHILE BLOOMING CAREFULLY WATER AS THE BLOOMS CAN BE DELICATE (THEY CAN BE HAND SIZE).THEY PREFER MORNING SUN OR BRIGHT INDIRECT OR FILTERED ALL DAY LIGHT. I HAVE NOTICED THAT A FEW HOURS OF DIRECT SUN (NOT WHEN SUPER HOT THOUGH) PROMOTES FLOWER GROWTH.

I found these cuttings for sale on E-bay:

http://cgi.ebay.com/ws/eBayISAPI.dll?ViewItem&item=170090097789&category=42211&ssPageName=WDVW&rd=1#ebayphotohosting

I hope I can eventually get mine to flower! I have to admit that I got rid of the mother plant, but not before making 5 plants from cuttings. My grandma's plant was huge, but in a 6 inch pot. It was soooo neglected, and it flowered religiously. When I bought her house I inherited the plant and it never flowered again. Hopefully its children will prosper!

Thanks for the info. I can now find out how to care for them (since I didn't inherit my grandma's luck with plants!)

I suspect neglect is part of the secret of getting these to bloom. That certainly seems part of the way I'd finally gotten my Xmas cactus to bloom.