The airport serving Kathmandu, Tribhuvan International Airport, is only international airport in Nepal. The airport has two terminals, an international, and a domestic terminal. If you are arriving into the international terminal, it is possible to take a taxi to the domestic terminal and take a domestic flight on the same day. However, you will have to check your luggage out upon arrival, and transfer it yourself to the domestic terminal.
The airport is also the major domestic hub, with flights connecting all major cities of Nepal.