The historical background is briefly described here:

http://en.wikipedia.org/wiki/Planck_constant#Origins

A more detailed historical background can be found in the biography of Albert Einstein, Subtle is the Lord, by Abraham Pais.

What perhaps is to be borne in mind is that h (or h-bar = h/2π) is the quantum of action. It follows that wherever one has an action, or action integral, in an exponent, one must also have 1/h, or 1/h-bar, in that exponent, so as to have a dimension-less quantity as the argument of the exponential function. This implies that in semi-classical and quantum descriptions of physical systems (notably in the path-integral formalism) h, or h-bar, is unavoidable (unless of course one adopts a system of units according to which h is identified with a dimensionless constant, such as h = 2π, which is the value used in the atomic units).

First of all it is Planck, not Plank. Secondly, the wikipedia entry skips most of the interesting history and in addition has a number of errors, or at least points of contention.

Briefly stated: Planck's constant came into existence through interpolation between two expressions valid in different regimes for the black body spectrum as a function of frequency. It must be the most fruitful interpolation ever.

Planck was very much interested in black body radiation. One of the reasons was that a major problem at the time (late 19th century) was understanding the second law of thermodynamics, and Planck believed that radiation damping could hold the key to irreversibility. At the time there were a number of experimental and theoretical results regarding the spectrum of a black body. There was Wien's law which correctly predicted the displacement of the maximum as a function of temperature and the Stephan-Boltzmann law ($T^4$ dependence of the energy). But it was incorrect at low frequencies for which there was Rayleigh-Jeans law, based on equipartition and the classical harmonic oscillator. That law, however, leads to the ultraviolet catastrophe. What Planck did was interpolate between those two results using some fundamental thermodynamics, basically $dS =dU/T +p/T dV$, from which $1/T = \left(\frac{\partial S}{\partial U}\right)_V$ follows. (It is about time RG implemented mathjax).

Planck solved both the Wien and the Rayleigh-Jeans expressions for $1/T$, differentiated once more, and interpolated between the two resulting expressions. Quoting Planck: "Pursuing this idea [of interpolation] I came to construct arbitrary expressions for the entropy which were more complicated than those of Wien, but acceptable. Among those expressions my attention was caught by $\frac{\partial^2 s}{\partial e^2} = \alpha/e(\beta +e)$, which comes closest to Wien's in simplicity and deserves to be further investigated." Fitting the black body spectrum to this interpolation fixes the constants $\alpha$ and $\beta$, which leads to what we now call $h$, and its value (skipping over some details here). Working backwards along lines set out by Boltzmann earlier, he then derived from the average energy in a mode, went back to the partition function $Q$ and showed that it can be written as an infinite sum over $e^{-n h \nu/k_BT}$ terms, in which we now recognize the energy levels of the quantum oscillator. Planck did not like that, even though it explained everything about the black body radiation, and he spent many years trying to find a more classical expression. A relevant quote from him regarding his derivation is: "a piece of mathematical jugglery without any correspondence to anything real in nature." His efforts were in vain, at the end he came to the following conclusion: "My vain efforts to incorporate the quantum of action somehow into the classical theory took several years and much work. Some of my colleagues have seen this as tragic. But I disagree...." Two other interesting books on this topic are Ingo Mueller's "History of Thermodynamics", and Thomas Kuhn's "Black Body Theory and the Quantum Discontinuity 1894--1912".

The second place where Planck's constant pops up is in the integration over phase space. This is a different, and unrelated problem, which is occasionally still being discussed in papers on the Gibbs paradox and particle indistinguishability. In the early 20th century this problem was referred to as "determining the chemical constant", and it is often suggested that quantum mechanics solved it. But not nearly everyone agrees. There are in fact two problems to be solved, one is the dependence on the number of particles, the second the value of a remaining free constant. In particular Ehrenfest wrote a number of papers on this topic for instance "Ableitung des Dissoziationsgleichgewichtes aus der Quantentheorie und darauf beruhende Berechnung der chemischen Konstanten", (1921). An interesting, and much later, contribution is by E.T. Jaynes ("The Gibbs Paradox", 1992) who shows that this constant is not necessarily equal to Planck's constant, since $e^2/c$ has the same dimensions, and is classical, and there is no particular reason why a constant 137.036 could also not be arrived at classically (if it is even needed). He also mentions that in quantum mechanics one still has the freedom to multiply $h$ by a constant. As far as I know this point was never addressed in later literature, and no attempts were made to give such a derivation. It is clear that a constant with dimensions of Js is needed, but as far as I know there is no derivation that it has to be $h$. Of course, starting from for instance the quantum particle in the box, in the high temperature limit (not the classical limit as Wikipedia suggests) you will get a correct expression for the entropy (Sackur--Tetrode), but that does not necessarily imply that the volume element in classical phase space should be $dxdp/h$.

Thanks "Gert Van der Zwan" for correcting spelling and above all so nice narration of whole thing from the beginning ....

Thanks "Behnam Farid" for very nice link,,,,,,,

With regards

I am very much interested in your question. As far as I know the value of Planck's constant `h' that we use everyday, which is 6.026 x 10-34 J.s, is found by using empirical methods only. There is no derivation of this value from a fundamental theory. At most we can say that it is the smallest possible quantum of action; but there is no theoretical argument from which we can derive this minimum possible value of action. Therefore the value of h must be known from experiments for the time being. Of course we hope that in future we will have a theory to explain its observed value.

Ofcourse the answer of Gert Van der Zwan above gives a nice and detailed historical account.

@Biswajoy, @Gert van der Zwan, there cannot be a something other than an empirical value. In fact as you may know in the proposed new version of the SI standard (https://en.wikipedia.org/wiki/Proposed_redefinition_of_SI_base_units), the value of h will be defined to be exactly 6.62606X×10−34joule second (J·s)  thereby defining  the kg (the X is still subject to change as the mass of the standard kg is compared more and more precisely with a Watt balance, which is then where all the empirical data is, and for which given that the standard kg is just a human artefact, there obviously is no alternative). The physically important statement is that there exists a "god given" (i.e. coordinate independent) volume measure on phase space, the Liouville measure, which is invariant under the dynamics of a Hamiltonian system (both statements are a direct consequence of the fact that phase space is a cotangent space and cotangent spaces have a "god given" symplectic form). Therefore, it makes sense to say that a region in phase space has a fundamental unit h of action. That in SI units h ≠ 1 Js is just as much historical bagage as the fact that c ≠ 1 m/s.  The other physically important question is why \alpha = 1/137.036.. as that is a dimensionless number.

@Gert van der Zwan,  See also this essay by Timothy Boyer who gives an argument that Planck could have introduced h on purely classical grounds as the scale of classical zero point energy of the electromagnetic field. http://arxiv.org/pdf/1301.6043v1.pdf

An idea which he seemed to have had since at least 1969: 

Boyer, Timothy H. "Derivation of the blackbody radiation spectrum without quantum assumptions." Physical Review 182.5 (1969): 1374.

http://journals.aps.org/pr/abstract/10.1103/PhysRev.182.1374