In a developed country, virtually everyone has access to a fairly high level of technology. In this case I am using the term technology to encompass a wide variety of inventions that make our lives easier or more comfortable, such as indoor plumbing, electricity, phones, indoor heating and air conditioning, modern medical treatments, and good transportation.
In developing countries, these things are available to the richer segment of the population, and the country is moving toward availability of these technologies to all. Undeveloped countries are ones where the majority of the population does not have access to such things.
There is not really much difference between "newly industrializing" and "developing." Really, they both mean the same thing but some people will think that the former term is less insulting than the latter. The term "developing" implies (to some people) that the country is not yet good enough.
A developed country is really just one that is fairly rich and has a diversified economy.