What is the meaning of percentile? I am confused by the term percentile. 
Once my teacher told me that percentile means the percentage with respect to the score of the highest achiever.

This means that if in a competition I got $80$ out of $100$ and the highest score in that competition was 90 out of 100 then my percentile would be $\frac{80}{90}*100=88.89$. 

So I got $80\%$ and $88.89$ percentile.

I was believing that my above concept was right.

But when I see the definition of percentile on Wikipedia then I got something new (but I don't understand this definition) and then I thought that what my teacher told me was wrong.

Kindly tell me if my teacher right or wrong.
If your memory is correct, your teacher was wrong.

The percentile tells you the value below which a certain percentage of the observations fall.

Thus, if the median score on the test was $80$, a score of $80$ would put you at the fiftieth percentile regardless of what the high score was.
First, consider the possibility that you misunderstood your teacher or that your words above do not quite capture what (s)he meant. That said, the "percentage with respect to the score of the highest achiever" has nothing to do with percentiles, and I am not sure there is a single word to express "I got within 89% of the best".

Percentiles (or, more generally, quantiles) are about relative positions compared to everyone else. It is not required that whatever you are looking at has numerical values, only that the values can be ordered. Taking an example in the spirit of @Nick's first link, consider a test where


60 people got a D
you and no one else got a C
39 people got a B
no-one got an A or worse than D.


How many % of a B are a C? However, one can still make statements in terms of quantiles, for example:


Everyone in the lower half got a D. (Since some in the upper half did so, too, the median grade is a D, too.)
Everyone in the highest (fourth) quartile got a B.
Everyone in the sixth decile got a D. (= those who scored better than the bottom 50%, but worse than the top 30%.)
Those in the 61st percentile got a C.
Those in the 62nd percentile got a B.


Thus, quantiles are best imagined as lining up everyone in (ascending) order and putting (ascendingly numbered) separators at n equidistant steps (n=100 for percentiles, 10 for deciles, 4 for quartiles)

Your percentile rank is then the number of the marker closest to you towards the "worse" side. (Note that percentiles make most sense if the considered set has >>100 elements. See the aforementioned link on how handle sets with &lt; 100 elements.)