The Dirac delta function
The Dirac Delta function is an idealized "spike" function. It's a mathematical abstraction,
useful to represent the extreme behavior
of a highly peaked functions.
Definition
A Dirac delta function, $\delta(x)$, has these properties: $$\delta(x)=\left\{\begin{array}{rl} 0;& \text{if }\ x \neq 0\\ \infty;&\text{if}\ x=0\end{array}\right.,$$ $$\int_{-\infty}^{+\infty} \delta(x) dx = 1.$$
Consider a Gaussian $$g(x) = C e^{-(x/w)^2}$$ with $C$ such that $$1=\int_{-\infty}^{+\infty}Ce^{-(x/w)^2}\,dx=C\sqrt{\pi} w.$$ So, $\Rightarrow C=1/w\sqrt{\pi}$.
You would be right if you guessed that this behaves exactly like a Dirac delta function in the limit... $$\lim_{w\to 0} C e^{-(x/w)^2} = \delta(x).$$
When a Dirac delta function meets another function
When we integrate the product of a Dirac delta function with any other function, since $\delta(x)$ is 0 everywhere except at the origin: $$\int_{-\infty}^{+\infty} f(x)\delta(x) dx = f(0)\int_{-\infty}^{+\infty}\delta(x) dx = f(0).$$
Now, $\delta(x-a)$ will be zero everywhere except at $a$, so we say this function "picks out" a particular value of another function when integrated together... $$\int_{-\infty}^{+\infty} f(x)\delta(x-a) dx = f(a).$$
For example: $$\int_{-\infty}^{+\infty} x^3\delta(x+1) dx = (-1)^3 = -1.$$
How sensitive is this integral to its limits??
How should we interpret... $\delta(kx)$?
Let's see how this behaves when integrated with another function. We'll need to change the variable of integration to $y=kx \rightarrow dy=k\,dx$:
$$\begineq\int_{-\infty}^{+\infty} f(x) \delta(kx) dx&=\int_{-\infty/k}^{+\infty/k} f(y/k) \delta(y) \frac{dy}{k}\\ &= f(0)\int_{-\infty/k}^{+\infty/k} \delta(y) \frac{1}{k}dy\\ &=\frac{f(0)}{|k|}.\endeq$$