Geographically, depending on who did the calculation, the center of Europe is somewhere between Southwestern Lithuania and the Northwestern edge of the Carpathian Basin [today in Slovakia and the Westernmost tip of Ukraine].

Historically, "Central Europe" used to be the Polish-Lithuanian Commonwealth, Prussia, and the Habsburg Empire (was earlier Austria and Kingdom of Hungary, became later Austria-Hungary). Culturally, this made some sense for the following reasons: Eastern border roughly the Eastern extent of Western Christianity (which meant very real political alliance systems and exchange beyond religion), these continental countries were off East of the dominant maritime nations making up Western (and Northern) Europe, and there was the Ottoman Empire to the Southeast. With German unification in the 19th century, all of Germany shited into "Central Europe".

To give some evidence, Central European Time (CET, the timezone today extending from Spain to Hungary, but originally only from Germany and on to Transsylvania and Galicia, areas then in Austria-Hungary but today in Romania resp. Ukraine) was established at the request of the Hungarian State Railways (see here).

In all the descendant countries, the region is called thus to this day. In the West, however, after the Iron Curtain descended across the continent, "Eastern Europe" became what was beyond.

After the fall of 'communism', there was confusion. Finding that the locals are confused and aren't that enthralled by their "Eastern Europe" terminology, some Westerners (especially those immigrating fro here...) had no trouble with the old terminology (see the Soros-funded Central European University). Then there are the new 'compromise terms', used on international fora to please us but apparently unknown to most Westerners: Central-Eastern Europe, Eastern-Central Europe (CEE/ECE), the same with "and".

But Western usage is confused also for reasons entirely unrelated to our sensitivities. What complicates the picture is

1. EU accession: no one says "Eastern EU", instead, the mostly Central European new members (and immigrants from there) became "Eastern Europeans".
   
2. the Russia question: Russophobes of various degrees wouldn't even see Russia as culturally European, or just ignore the ex-USSR-minus-Baltics - so the Eastern border of the EU is really the cultural limit of Europe for them.
   
3. Poemless would also want me to add to the previous a parallel trend among Central European Russophobes, insisting on differentiating themselves as Central European to distinguish from the 'russkies'. (Though I honestly don't know how much, if any, role that plays.)