Could you please give an intuitive definition of chemical potential? Could you please give an intuitive definition of chemical potential? It seems that it is an extremely important notion of physics but definitions are really vague.
 A: The chemical potential can also be viewed as a Lagrange multiplier related to the constraint that the number of particles of a closed system is fixed.
To understand this, consider the example of a gas made of $N$ electrons. The probability of an eigenstate of energy $E$ be occupied is given by the Fermi-Dirac distribution
$$n_F(E,\beta)=\frac{1}{e^{\beta(E-\mu)}+1},\tag1$$
where $\beta$ is the inverse of temperature. This equation can very well be understood as the definition of the chemical potential. If we sum over all momentum eingestate $\vec k$ and multiply the result by two to account for the two spin states we obtain the total number of particles,
$$N=2\sum_{\vec k}n_F(E(\vec k),\beta).\tag2$$
Since in general $n_F$ changes with temperature, as in Eq. (1), and yet $N$ has to be kept fixed, this intuition about the chemical potential is that it is whatever parameter $\mu$ defined by (1) which satisfies the constraint (2). 
A: You say that the definitions are vague, but $\mu_i=\left(\frac{\partial U}{\partial N_i}\right)_{S,V,N_{j\neq i}}=\left(\frac{\partial G}{\partial N_i}\right)_{P,T,N_{j\neq i}}$ is precise.
However, it may be helpful to use an analogy to obtain an intuitive definition. I'm sure you're familiar with the fact that systems tend to evolve to reduce gradients. Any such spontaneous change involves two conjugate parameters: a generalized force (corresponding to a gradient in some field, such as a pressure difference) and a generalized displacement (corresponding to the flow, such as a change in volume). The product of the two conjugate variables has units of energy.
In heat transfer, for example, a temperature gradient causes a spontaneous flow of energy. The "stuff" that is transferred is entropy. Thus, we obtain the differential term $dU = T\,dS$.
A pressure gradient drives a change in volume: $dU = -P\,dV$.
What would cause spontaneous movement of matter? In this case, the driving force is a gradient in the chemical potential of a material $i$: $dU = \mu_i\,dN_i$.
I'm sure you're also familiar with the concept that changes in concentration drive material transport or diffusion. This is only an approximation. It fails to explain why oil and water separate, for example. (Or why any two mixed materials would separate.) The chemical potential is like an augmented concentration that also incorporates bonding between materials (as well as concentration). It is the true arbiter of how matter will move.
A: The physical interpretation of chemical potential that works best for me is the "private" Gibbs free energy per mole of each individual chemical species in a mixture, so that the total free energy of the mixture is just the number of moles of each species times the chemical potential of that species, summed over all the species.
A: You can say that its the energy needed to remove a particle from a many-body system.
More formally
$$F(N+1) -F(N)=\mu$$
where $F$ is the free energy of the system and $N$ is the particle number.
A: While each of the earlier answers gives insight into the meaning of chemical potential, personally the Wikiperdia entry https://en.wikipedia.org/wiki/Chemical_potential gives me the best sense of how to read chemical potential: as well as distinguishing "electrochemical potential" from chemical potential, it also interprets its meaning in a wider variety of physical systems. 
