So sorry, TK, best of luck to you and your mother!
As for your confusions, I'm willing to bet that the two names that were given were small cell and non-small cell lung cancer. Basically, the chemo that's used for those cancer's are different for each one, so they need to do a biopsy to see which one she has. Small cell cancers generally are more aggressive and carry poorer prognoses.
As for 3b, that indicates the stage of the cancer. It's basically the doctor's guide to how much the cancer has progressed. Stage IIIb means that the cancer has not metastasized from what they can tell, but either it has ingrained itself into surrounding structures (like the esophagus) or cells from the tumor have gone into lymph nodes in the region.
Hope this clarifies some things
Again, best of luck to you and your family.