Welcome aboard Orchidopile. Those orchids belong to the Dendrobium genus.
The purple one generally falls under the "phalaenopsis-type" dendrobium because the flowers resemble phalaenopsis flowers.
The white one I think is closer to an antler-type dendrobium because of the flower's shape. I'm not 100% sure about that though.
However, without a proper tag it will be next to impossible to identify them since dendrobiums have been hybridized up to the wazoo. There are probably hundreds of unregistered hybrids around the world by now.
In terms of crossing the orchids, there is a specific section here on the boards that deal with orchid propagation. Check out the threads there as there are plenty of informative discussions there regarding orchid breeding.