You have a Guzmania. There are many species and hybrids so id is difficult. Could be lingulata.
You can search through the Photo Index at http://fcbs.org/ for a better guess.
Rain water is best (certainly in the cup), but for watering in the pot, tap water is usually ok.
Most Bromeliads are "determinate," i.e. each "branch" only flowers at its own terminal bud. (This bud is very dominant and supresses dormant buds below it.) Once this happens (as yours has), then the plant will produce new branches lower on the stem (these are the pups or offsets).
The brown leaves are dead (just like old pine needles) - remove them and make room for the new pups.
Critical for you will be maintaining humidity. These are rainforest plants and expect (in nature) to be rained on just about every day. This doesn't mean you can't succeed, just that you will have a bit more challenge than someone in say Florida. If your humidity is low, you may experience the leaves "quilling," i.e. growing in a tight tube sticking together. This usually results in that branch dying off.
You should have plenty of light in Arizona, although Guzmanias are not high light plants (certainly in the Summer). Through the Winter months you can give them full sun if you ease them into it.
Dave.
