Why do prices ending in 99, e.g., 0.99, 199 or 1999.9 convince consumers to buy? Why are people more likely to buy articles with a price of \$199, rather than the same (or let's say a slightly better) article for \$200?

On which human brain centre is this advertising focused? Is this some kind of manipulation?
An interesting article relating to your first question is the "Analyzing the Psychology of Pricing", where they suggest that consumers don't really notice any change in units of 10, e.g. \$54 vs \$59.99 etc.

In answer to the 2nd of your queries about what part of the brain is affected, here is some research that I found.

According to the article "Marketing actions can modulate neural
representations of experienced pleasantness" (Plassman et al. 2007), very little is known about the neural responses to marketing.  However, they tested the hypothesis that 


  marketing
  actions, such as changes in the price of a product, can affect neural
  representations of experienced pleasantness.


In another of that author's papers "Branding the brain: A critical review and outlook" (which includes diagrams of the parts of the brain affected), he bases the process on the parts of the brain that control


  (1) representation and attention, (2) predicted value, (3) experienced
  value, and (4) remembered value and learning.


This is also something that many in marketing are actively pursuing, according to "Neuroeconomics: Studying Brain Responses Gives Marketers Increased Ability to Predict How People Make Decisions".