There is not a great deal of public transport in Namibia. This fact can quite simply be attributed to the fact that the country does not have a great deal of public (a population of around 3 million people live in a vast area).
This does make getting around fairly difficult and conventional wisdom is to either join an organised Namibian safari or rent a car and enjoy Namibia on a self drive holiday. However if you are willing to take your chances and have plenty of time available then it is possible to get around the country using a combination of public transport and hitchhiking.
The most popular form of transport between towns is the Mini-Bus. This is typically a 15 seater vehicle. These do not run according to a timetable and will leave when they are full, if they do not full up today you may have to return tomorrow to try again. The good news is that they are a very cheap way to travel and you are likely to meet some interesting locals and pick up a few stories about the near accidents that occurred on your harrowing journey. As a rule of thumb these mini-busses link anywhere that is on a tarred road (having a tarred road running to your town means it is of some importance!)
In towns, there are no real bus services, however in Windhoek the function of a bus service is operated by local taxis. These are generally sedan cars, in varying states of repair, which operate along designated routes - they operate on a shared basis so do not be supervised when your driver suddenly stops to pick up extra fares. There are also traditional metered taxis, generally thought to be safer and exclusively for your private use, but lets be honest where is the fun in that!
The one major issue with public transport is that although it can get you to towns around the country the main reason for visiting is to see the splendour of Namibia, and Namibian towns are not particularly splendorous. (I once wrote something uncomplimentary about Swakopmund and actually got hate mail from local residents - so I am excluding Swakopmund from this comment, it is truly a splendorous place!). Anyway back to the original point if you want to see Etosha Park, the Fish River Canyon or Sossusvlei it is not going to be possible to use public transport to get there.
"I have opinions of my own -- strong opinions -- but I don't always agree with them.", George Bush