Learn English – difference between the words “concious” and “conscious” (and is the former even a word)

Originally, everyone spelled it centre, but because of Noah Webster's spelling reforms, people in the US started spelling it center, particularly in the last century. Although the revised spelling center has been adopted internationally to varying extents, centre is still more popular in most regions. But regardless of how you spell it, it's the same word, so it has the same meaning either way.

To get a clearer picture of which spellings are used in which countries, I turned to the Corpus of Global Web-Based English (GloWbE). After searching for both terms, I took the raw data and created a chart of percentages using the formula center / (center + centre), which I sorted in descending order. Here's what I found:

Percent of the time center is used rather than centre, by region

  United States 92.2%
    Philippines 87.5%
       Pakistan 54.0%
      Hong Kong 51.8%
     Bangladesh 51.2%
          India 43.8%
       Tanzania 42.5%
      Singapore 40.7%
          Kenya 40.3%
        Jamaica 40.3%
        Nigeria 39.1%
       Malaysia 35.1%
          Ghana 34.4%
      Sri Lanka 33.8%
         Canada 33.0%
    New Zealand 24.2%
        Ireland 20.1%
   South Africa 19.8%
  Great Britain 19.4%
      Australia 18.1%

As you can see, center is more popular in the US, while centre is more popular in the UK. Many regions fall closer to the center of the spectrum, with India favoring centre only somewhat, and Hong Kong using both spellings in roughly equal proportions. Overall, centre is more popular in international English.

Which should you use? Well, if you're required to follow a style guide, use whichever spelling it suggests. If you have a choice, and you're writing in a region like the US or UK that strongly prefers one spelling, I suggest you use that spelling. (I don't think either spelling is ever "incorrect", but it's possible that the person grading your test might disagree with me!) Otherwise, you can spell it however you like.

Finally, I'd like to add one note about the percentages above. The tendencies at the top and bottom ends of the chart are likely to be stronger than the percentages indicate, primarily because spellings are usually respected in proper nouns regardless of region. If you talk about the Capital Centre, for example, you're likely to spell the word centre rather than center regardless of which region you're in. But since most uses of this word are not in proper nouns, the percentages should be roughly accurate, particularly toward the center of the list; it should be enough for you to decide how to spell the word.

Authorative is a misspelling.

I looked up "authorative" in as many online dictionaries as I could find (well, actually, I had OneLook Dictionary Search do it for me, but same effect), and found only one result, in Wordnik. However, that one result does not actually contain a definition, just some examples of use. All of the examples clearly meant to say "authoritative".

Google is immaterial here: people misspell things on the internet. If you mean to say that something is

Having or arising from authority; official

then you need to write authoritative.

(Note that the reason for this is that this is an adjectival form of authority, not of author.)