Has there ever been an application of dividing by $0$? Regarding the expression $a/0$, according to Wikipedia: 

In ordinary arithmetic, the expression has no meaning, as there is no number which, multiplied by $0$, gives $a$ (assuming $a\not= 0$), and so division by zero is undefined.

Is there some other kind of mathematics that is not "ordinary", where the expression $a/0$ has meaning? Or is the word "ordinary" being used superfluously in the quoted statement?
Is there any abstract application of $a/0$?
 A: You said you wanted an application.  Inspired by the example from Exceptional Floating Point, consider the parallel resistance formula:$$
    R_{total} = \frac{1}{\frac{1}{R_1} + \frac{1}{R_2}}
$$This formula tells you the effective electrical resistance of a path when the current can choose two routes to take.
Let's pretend that $R_1=0$.  Then we have:$$
    R_{total}=\frac{1}{\frac{1}{0}+\frac{1}{R_2}}=\frac{1}{\infty+\frac{1}{R_2}}=\frac{1}{\infty}=0
$$The resistance being zero is indeed the correct answer; all current flows along the single wire that has no resistance.
Naturally, you need to make appropriate definitions for arithmetic on $\infty$ (i.e., use the projective reals).  For well-behaved applications like this, that's fairly straightforward.
A: Theory of relativity.
$$m = \frac {m_0}{\sqrt{1 – (\frac vc)^2}}$$
This means that as your velocity (speed) increases, and gets closer and closer to the speed of light, your mass increases (therefore, mass is related to velocity). It also proves, that it is impossible to travel faster than the speed of light. If an object were to do that, its mass would reach infinity, and that is impossible, so travelling faster than the speed of light is impossible.
A: Though not actually correct, I've seen it used to "trick" people with a proof that 1=0

Consider two non-zero numbers x and y such that
$x = y$
Then $x^2 = xy$
Subtract the same thing from both sides:
$x^2 - y^2 = xy - y^2$
Dividing by $(x-y)$, obtain
$x + y = y$
Since $x = y$, we see that
$2 y = y$
Thus $2 = 1$, since we started with y nonzero.
Subtracting 1 from both sides,
$1 = 0$

Su, Francis E., et al. "One Equals Zero!." Math Fun Facts. http://www.math.hmc.edu/funfacts
A: In complex analysis, we talk about the value of a function at infinity. To evaluate $f(z)$ at infinity, compute $f(1/z)$ then plug in $0$. This allows us to talk about things like the order of zeros and poles at infinity.
Example: $$f(z) = \frac{az+b}{cz+d}$$ with $ad-bc \neq 0$ and say $a,c \neq 0$. $$f(1/z) = \frac{\frac{a}{z}+b}{\frac{c}{z}+d} = \frac{a+bz}{c+dz}$$ So $f(\infty) = \frac{a}{c}$.
A: In two dimensional space, if we want a vector corresponding to a given angle $t$, we can do:
$(x, y) = ( \cos(t), \sin(t) )$
But what about the reverse?  We have a vector, and want to know the angle.  A common solution is to use arctan:
$t = \arctan( \frac{y}{x} )$
Here $\frac{y}{x}$ represents the gradient of the vector. So it has a meaningful value even when $x = 0$.

Trivia: Many program languages will fail to perform that calculation, because division by zero will produce an error.  They usual address this by offering an alternative function which takes two arguments: atan2(y, x).
But a notable exception is Javascript, which has the concept of +Infinity and -Infinity as numbers, and can perform this calculation even when x = 0.
$ node
> Math.atan( 1 / 0 )
1.5707963267948966           (Pi/2)
> Math.atan( -1 / 0 )
-1.5707963267948966          (-Pi/2)

Another language which can perform this calculation is Haskell.

Inspired by Qiaochu Yuan's comment.
A: The algebraic structure "wheel" is an algebra with division by zero.  The one point compactification of the complex plane into the Riemann sphere almost produces a wheel (one still needs to adjoin the element $0/0$).
A: I gave an answer to a similar question some time ago here.
To put it briefly, if you are ever solving an equation in one variable by just addition, multiplication, subtraction, and division, and you end up with a contradiction, then you can conclude with certainty that the one time you divided both sides by an expression involving a variable, the expression you divided by was actually $0$ (because having multiple answers can only happen if you had a $\frac 00$).
Also, when computing limits, if you get a $\frac a0$ for $a\neq0$, you can directly conclude the limit does not exist (as a finite number).
A: This may not be what you would consider an "application', but for computer floating point arithmetic, division by zero is useful for setting up three special values: 1.0/0.0 gives +inf, which is a valid floating point value satisfying the usual extended number line properties. -1.0/0.0 gives -inf, and my all-time favorite is 0.0/0.0, which gives NaN (not-a-number). In spite of the name, NaNs are also valid floating point numbers in the IEEE 754 standard, which almost all modern computers implement. A few languages like MATLAB let you directly specify those (e.g., a statement like x = inf is allowed) but others do not allow this - requiring the divide by zero syntax.
Possibly irrelevant to you but NaNs are useful in some numerical computing applications and highly useful in debugging some codes. The plural "NaNs" is valid because there are many of them: at least $2^{51}$ of them in 64-bit arithmetic. If you are interested, look for papers written by W. Kahan (father and grand inquisitor for the IEEE 754 standard). 
A: Yes, in projective geometry or hyperbolic geometry for example, you can see applications or geometric entities that are $\frac{a}{0}$ or just $ \infty$ . Generally, non-euclidean spaces have such type of entities or applications.
