Understanding the definition of omnibus tests From Wikipedia

Omnibus tests ... test whether the explained variance in a set of
  data is significantly greater than the unexplained variance,
  overall. 
...
Omnibus test as a statistical test is implemented on an overall
  hypothesis that tends to find general significance between parameters'
  variance, while examining parameters of the same type, such as:
  Hypotheses regarding equality vs. inequality between k expectancies
  $µ_1=µ_2=…=µ_k$ vs. $µ_1≠µ_2≠...≠ µ_k$ in Analysis Of Variance(ANOVA)
  ;

I was wondering what it means by a variance  being explained or unexplained?
How shall I understand what an Omnibus test is?
What is a set that is not omnibus?
Thanks and regards!

Added: I was wondering if the following likelihood ratio test for the global null is an omnibus test, quoted from Casella and Berger's Statistical Inference?

...

 A: I wouldn't look for a rigorous definition of omnibus test. It seems typically used for overall tests with wide scope, packing several tests into one. 
Other terms used with similar import are portmanteau statistic and factotum statistic. 
Over a century or more, there have been all sorts of fashions over terminology, including statisticians reaching for their Latin and Greek (ancillary, histogram, chi-square, heteroscedasticity), statisticians reaching for their thesaurus (as here), statisticians naming tests after their teachers or friends, ideally in pairs (Mann-Whitney, Kruskal-Wallis), and statisticians eager to show off their homespun sides (jackknife, bootstrap). Even the words that look familiar had to be invented (average, mode, regression). 
A: 
I was wondering what it means by a variance being explained or
  unexplained?

It the context of ANOVA it means the variance "explained" by group membership and the variance that remains unexplained.  To understand this in detail you have to really look at the equations.  I'll try to explain it anyway without introducing too many equations.  In the case of a one-way ANOVA each value observed can be thought of as being composed of three sources of variance, the grand mean, the group mean deviation from the grand mean, and error... $x = \bar{\bar{x}} + (\bar{\bar{x}} - \bar{x}_k) + e$.  If you assume there are no group differences then all $\bar{x}_k = \bar{\bar{x}}$ therefore, by estimating $\bar{x}_k$ you haven't 'explained' very little or no extra variance.  Imagine instead that the null hypothesis is false and you go ahead and estimated the deviations of the group means from the grand means.  If you then adjust each score by the deviation of the group to which it belongs and then recalculated the variance of the scores, you will find that the variance is smaller than it was before.  That reduction in variance is the variance that you 'explained' by estimating means for each group. 

How shall I understand what an omnibus test is?

Tests are referred to as omnibus if after rejecting the null hypothesis you do not know where the differences assessed by the statistical test are.  In the case of F tests they are omnibus when there is more than one df in the numerator (3 or more groups) it is omnibus.  In the case of Chi-square tests, when there is more than one df it is omnibus.

What is a set that is not omnibus?

Comparisons between two groups such as those that happen in the cases detailed above F with 1 df in the numerator and Chi Square with 1 df.
