Did you read the Wikipedia article yet? The newsreaders you hear on Deutsche Welle all speak standard German or at least they are supposed to. As a learner of German residing abroad, you should aim for "standard German" pronunciation. Most movies made in Germany have their actors speak standard German except when dialect is indicated by the script. Your tapes in your language lab are also in standard German.
If you can make your question more specific, please do so.
(after editing of Question:)
Well, there is a widespread notion that Sächsisch as spoken in and around Meißen was considered the most refined form of spoken German, that later some other regional variant (perhaps as spoken in Prague) took its place, and that the standard German spoken by educated speakers in and around Hannover is setting the tone today. Coupled with this is the notion that this "Hannover standard German" is not really a dialect but a form of German marked by the absence of dialectal color (a notion that I admit to sharing).
I think the real history of how and why "standard German" was constructed and how it changed over time is more complex. Yesterday I tried to look for a good historical account on the Web but found none. (Suggestions welcome!) If you listen to old newsreels from the '40s and earlier, the standard German spoken by newsreaders sounds very different from what we hear today in major news broadcasts.
Why and how did a northern/northernish German pronunciation become the touchstone for "correct" or "good" spoken German? I think a lot may have had to do with the post-war years, when Germans looked to the British occupation forces and the BBC for guidance on matters of journalism. Germany's premier television newscast is the Tagesschau and it has had only five Senior Newsreaders since 1964. Of these, three -- Köpcke, Veigel and Hofer -- grew up in areas that later happened to be in the British zone of occupation. Berghoff is a Berliner, Brauner a refugee from Lower Silesia in the east. No Senior Newsreader at Tagesschau has ever come from a "southern" state (Hessen, Rheinland-Pfalz, Saarland, Baden-Württemberg, Bayern).
The longest-serving Tagesschau newsreader was Karl-Heinz Köpcke (started in 1959, Senior Newsreader from 1964 to 1987), a native of Hamburg. A modest, reserved man, he had great influence not only on the direction in which "standard German" -- which, as you may have gathered by now, is a constructed, "synthetic" language although its chief architect was "contingency" -- moved but also on the image of German "maleness".
A point that hasn't been mentioned yet is that when people speak Hochdeutsch, i.e., standard German, they don't necessarily sound like a Tagesschau newsreader. In fact, 99.99% of the time you can still hear quite clearly where they come from! The difference is that when speaking Hochdeutsch, they (try to) use only words found in dictionaries (excluding dialect words) and grammar as taught to them in school. Bavarians have never been shy about laying on their regional accent when speaking Hochdeutsch; Swabians, on the other hand, are a bit self-conscious about it. Recently they turned their embarrassment to their advantage when they chose the slogan "Wir können alles außer Hochdeutsch" for a regional marketing campaign.
Getting back to Hannover and vicinity, I don't necessarily agree that people there speak the "best" (or clearest, or whatever positive adjective you choose) German. People like Gerhard Schröder (German chancellor from 1998 to 2005), Frank-Walter Steinmeier (Foreign Minister from 2005 to 2009) or Sigmar Gabriel (chairman of Germany's Social Democratic Party) have so much macho metal in their voices, I'm surprised they don't run a steelmill on the side. I find this metallicity disagreeable to listen to.
The best German speaker I have ever heard was a co-worker of mine in a heavy-machinery moving company. This man, barely educated past tenth grade, spoke with a naturalness and clarity that I have not heard before or since. For what it's worth, he had grown up in Wuppertal, the largest city of the Bergisches Land, which is about halfway between Cologne and Dortmund.
Regarding the two other major German-speaking countries: Schweizerdeutsch (Schwyzerdüütsch) is not considered a dialect but a language in its own right. However, newsreaders on Swiss TV sound just like their counterparts on Tagesschau. In Austria, I believe that the most refined pronunciation is called "Schönbrunn-Deutsch" and to my untrained ear, some of their TV newsreaders seem to employ this very Austrian-sounding pronunciation. However, you'll have to ask a Swiss or an Austrian for reliable information on these details.
This interview with Prof. Karl-Heinz Göttert in WELT Online is absolutely fantastic. Can't resist inserting a few quotes:
Wenn in Schwaben ein entsprechender zentraler Hof entstanden wäre,
wäre dasselbe passiert wie in Frankreich und England: Aus den
Dialekten der Île de France und von London wurde die Nationalsprache,
und so wäre vielleicht der schwäbische Dialekt zum Deutschen
Whew (wipes sweat off brow)
Die Leute, die festgelegt haben wie das Deutsche korrekt ausgesprochen
werden soll, waren alle Niederdeutsche. Es ist eine ganze Kette, wie
eine Verschwörung. Bis hin zu Herrn Siebs, der mit seinem Buch "Die
deutsche Bühnenaussprache" für die Lautung im 19. Jahrhundert so
entscheidend geworden ist wie Konrad Duden für die Rechtschreibung.
Der Siebs konnte von Haus aus überhaupt kein Deutsch. Der ist in
Bremen geboren und auf einer friesischen Insel aufgewachsen. Und so
ein Mensch hat die Aussprache des Hochdeutschen festgelegt. Deshalb
beginnt sagen heute mit einem weichen S. Der Bayer spricht es ja mit
scharfem S: "Ssag amoi..." Niederdeutsch ist auch die Aussprache ich
statt ik in der letzten Silbe von Wörter wie König oder wahrhaftig.
What this second quote illustrates is the contingency and artificiality of standard German pronunciation. Indeed, who the hell was Mr. Siebs to tell everyone how German should be pronounced "correctly"? There is no "objectively" correct pronunciation and there never has been. One may presume that there is a "de facto" correct pronununciation and that the synthetic standard German spoken by Mr. Jan Hofer, the current Senior Newsreader of Tagesschau, is its embodiment. It is an arbitrary choice but, given the influence traditionally exerted by Senior Newsreaders, a sensible one. (Does Hofer conclude "König" or "wahrhaftig" with an -ich or an -ik sound? I don't know, and it doesn't matter, because both choices are widespread and therefore acceptable standard German pronunciation -- regardless of what some silly book says.)
For a learner of German residing abroad, there are few opportunities to hear people speak standard German in daily intercourse, so as mentioned previously, there are the Deutsche Welle and other news broadcasts, cultural "magazine" shows on radio, practice tapes in the language lab, pop songs, etc. etc. More than 99 percent of the time, the speakers are using standard German pronunciation, albeit with minor regional coloration and with a few idiosyncrasies. The question of "which German dialect is closest to standard German" is a bit of a red herring, as you won't know where somebody heard on the radio grew up. In any case, as I've tried to show above, the local language spoken by people in and around Hannover, while indeed generally thought to be closest to "standard German pronunciation", isn't necessarily the most pleasant-sounding.