Nepal is one of the more straightforward visas in the region: you can get it on arrival, the prices are published, and the tiers are simple. The parts worth reading in advance are the extension arithmetic and the list of people who pay nothing.
The three tiers
Tourist visa on arrival costs USD 30 for 15 days, USD 50 for 30 days, or USD 125 for 90 days.
Pick deliberately. Extending later is not free, and the gap between tiers is often smaller than the cost of extending into the next one.
Extensions, and the arithmetic that catches people
Extensions cost $3 per day, with a minimum of 15 days. So the smallest extension you can buy is $45, whether you need fifteen days or two.
Overstay carries a $5 per day late fine on top of the extension cost. That is charged in addition, not instead.
Total stay on a tourist visa can be extended up to a maximum within a single visa year, so extending is not indefinite.
Who pays nothing
Nepal publishes a gratis list. It includes children below 10 years, SAARC citizens except Afghanistan for up to 30 days on a first visit in a given visa year, NRN card holders, and Chinese nationals.
Afghan citizens can obtain a gratis visa on the recommendation of the Department of Immigration, which is a different route rather than an automatic one.
If you are American
There is a reciprocity arrangement worth knowing about: US citizens and dependents can obtain a five year visa allowing 180 days of stay per year, for $160, issued by the Department of Immigration.
For anyone visiting Nepal repeatedly, that is a materially different proposition from buying a 90 day visa each time.
Getting the visa before you fly
If you obtain a tourist visa from a Nepali diplomatic mission abroad rather than on arrival, you must enter Nepal within six months of the issue date. The clock starts when it is issued, not when you land.
Transit
Transiting through Nepal permits a maximum 24-hour stay. If your connection is longer than that, you are not transiting as far as immigration is concerned, and you need an actual tourist visa.
Applying online
Visa applications, including for the categories above, go through the official Nepaliport system rather than a third-party site.