As Juan Weisz mentioned, the Heaviside function is discontinuous hence not differentiable. However, the Dirac delta is the distributional derivative of Heaviside step function. To read more about this one can see .

The Dirac delta over the real number axis is zero everywhere except at zero

when it is infinite.

The Kronecker delta 0ver integers is zero everywhere except at zero when it is equal to 1

The step is 1 over the positive real axis, but zero over the negative real axis.

Despite certain criticism of pure mathematicians about the use of such singular

functions all three are fundamental and of heavy use in applied mathematics.

As Juan Weisz mentioned, the Heaviside function is discontinuous hence not differentiable. However, the Dirac delta is the distributional derivative of Heaviside step function. To read more about this one can see .

In 1958 a small but excellent book has been printed:

M.J. Lighthill, Introduction to Fourier Analysis and Generalised Functions, Cambridge University Press.

Read the book and all your problems with generalised functions are solved.

The Heaviside function can be applied to modeling impulses and piecewise-defined function.

Some real applications can be found in my tasks in connections to the attitude dynamics of spacecraft, e.g.:

http://www.doroshin.ssau.ru/pub/AA-Dor-2014_accept.pdf 

http://www.iaeng.org/publication/WCE2009/WCE2009_pp1544-1549.pdf

Nobody mentioned yet the Kronecker delta.  Formally, it is a function of two integer arguments (usually positive or zero), equal to 1 if both arguments are equal to each other and zero otherwise.  It is probably most often found in situations where the so called Einstein notation is used, i.e. explicit summation symbol is omitted (summation limits are assumed obvious) and the summation itself should be performed using repeated indices.  You may also find it in linear algebra expressions: for example identity matrix is often written as having entries equal to Kronecker delta.

E.g. nuclear power stations are such real life applications of physics based on the mathematical usage of the delta (generalized)-functions. In the same spirit every high-tech instrument/gadget is a real life application of the Dirac delta, as well as of all other deltas (and of the relativity theory, of quantum mechanics, of the biotechnology etc.).

Details about these notions cannot be presented briefly without a suitably developed mathematical formalism, cf.  suggestions given in some answers above.

In particular it is inapproprite to say that Dirac's delta is a function. Correctly is to see it as an example covered by the extended notion of the integral functional. For instance, as a limit of such functionals represented by functions, wheras the limit functional is not represented by any function,  cf. also the book

by J.Mikusiński and R. Sikorski: The elementary theory of distributions, Tom 1, PWN, Warsaw 1957:

https://books.google.pl/books?redir_esc=y&hl=pl&id=iQQ_AQAAIAAJ&focus=searchwithinvolume&q=delta

Regards