Hi Carla,
This sounds like a pretty typical case of an upper respiratory infection or URI. The most common pathogens are Herpes virus and Chlamydia. It is a contagious disease and most cats will have the virus in their nasal cavity, only showing signs when stressed. The virus will run it's course over 7-10 days, there is also an incubation period, hence the delay before the other cat became clinical. When I see these cases I usually put them on an antibiotic (Clavamox) to help speed up recovery and prevent any secondary infections. It's not always needed, however.
If eating/ drinking and otherwise normal, you can probably just let it run it's course. Full recovery is expected.
Hope that helps, let me know if you have any other questions.
Cat Veterinarian
DVM, Emergency Veterinarian; BS (Physiology) Michigan State Univ