New Growth, Newbie cactus person, What and How??

I bought this at walmart about 3 years ago. It had "yellow fingers" at the top point.
This past winter the fingers started to fall off, I thought it was going to die on me.

It lost all it's yellow fingers and then a new growth popped out of the side.
I was happy I didnt kill it. (I'm a newbie with cactus).

Should I cut this growth off, (it is getting large fast). Or should I leave it on? If left on, will it grow more grow?? Or if I cut it off, will the mother plant produce more???

By the way, could anyone tell me the name of this type of cactus? I would love to read more about it.

Thanks for your help!

The top part was probably a 'grafted' cactus. The cactus that was grafted on top won't grow back.
The bottom part, the name of the species eludes me right now but, yes it will continue to grow new branches. It *might* bloom at night. They like a little more humidity than where I live. It needs frost and cold protection, and protection from direct hot sun. It comes from a tropical jungle enviroment.

It's probably 'Selenicereus grandiflorus'
http://images.google.com/images?hl=en&lr=&q=%22Selenicereus+grandiflorus%22&btnG=Search

I think it's probably a Hylocereus of some sort. It's a night blooming cactus from the tropics. In southern Cal, they grow them for the fruits which are the dragon fruits. They also make great grafting stocks.

Agree, Hylocereus sounds good too.