In the canonical ensemble, giving up a fixed energy bought us a fixed temperature. Here we go one step further and let the particles themselves come and go.
The wall in the figure has an opening, and particles pass through it. The number N inside the system is no longer fixed. What is fixed instead is how crowded it is outside.
Why set things up this way? Because for some problems it is easier to solve. Quantum statistics is exactly such a case: with the particle number held fixed, the awkward business of counting identical particles without telling them apart remains. This setting makes that awkwardness disappear entirely.
Just as temperature governed the exchange of energy, something is needed to govern the exchange of particles. That is the chemical potential \mu.
Its meaning is plain: the energy it takes to add one particle. Strictly, the energy to add one without changing the entropy. The ladder in the figure is the total energy as particles are added one at a time, and the height of one step is \mu.
Just as heat flowed from the higher temperature to the lower, particles flow from the higher chemical potential to the lower. Where the two become equal is equilibrium with respect to the exchange of particles.
The canonical weight was e^{-\beta E}. Following the same argument for the coming and going of particles, the weight becomes e^{-\beta(E - \mu N)}. This is the grand canonical distribution.
The point is the -\mu N added in the exponent. Add one particle and the weight of that state is multiplied by e^{\beta\mu}. The larger \mu is, the more likely the states with many particles become.
The grid in the figure lays out the states by energy and by particle number together. The depth of colour is the weight, darkening or fading as you go across and down. Move \mu and the dark region shifts.
Adding the weight over every state gives this setting's partition function. It is written \Xi = \displaystyle\sum_{i} e^{-\beta(E_i - \mu N_i)} and called the grand partition function.
It is used just as the canonical one was. Differentiate \log \Xi in \beta for the energy, and in \mu for the mean number of particles. The figure sets that out one line at a time.
So nothing about the practice changes. One more thing has been added to the sum, and the machinery beyond it is the same.
Here is the reason for choosing this setting in the first place. If the particles do not interact, a state of the whole system is settled by how many sit in each level.
And since the particle number is not fixed, each level may be filled independently of the rest. Had it been fixed, the tie that the total must come to N would remain, and the levels could not be separated.
So the grand partition function splits into a product of small sums, one for each level. The figure is that splitting, and all that remains is to solve one level. That one level is the Fermi and Bose distributions of the next piece.