I'm going to borrow parts of a recent post that I wrote in another thread to setup one aspect of the problem.
Consider for concreteness the theory of a single scalar field with potential [itex]V\left ( \Phi \right )[/itex]. The action is given by
[tex]S= \int d^{4}x\sqrt{-g}\left ( \frac{1}{2}g^{uv}\partial _{u}\Phi \partial _{v}\Phi -V\left ( \Phi \right )\right )[/tex]
The corresponding energy momentum tensor is computed as:
[tex]T_{uv}= \frac{1}{2}\partial _{u}\Phi \partial _{v}\Phi + \frac{1}{2}\left (g^{\alpha \beta }\partial _{\alpha }\Phi \partial _{\beta }\Phi \right ) g_{uv} - V\left(\Phi\right) g_{uv}[/tex]
The lowest energy density configuration if it exists is obtained when both the kinetic and gradient terms vanishes, which gives us our definition for vacuum energy and implies for this particular theory that:
[tex]T_{uv}^{vac}\equiv -\rho _{vac} g_{uv} =-V\left(\Phi_{0}\right) g_{uv}[/tex]
Where [itex]\Phi _{0}[/itex] is the value that minimizes the potential. Note that this is not necessarily zero. More generally you can argue by lorentz invariance that the form for vacuum energy is unique and fixed exactly as above.
Now, when you introduce quantum corrections, you can take my word for it, or you can perform the calculation yourself and you will get the first corrections that look like:
[tex]T_{uv}^{quantvac}\equiv -\rho _{quantvac} g_{uv} =-V\left(\Phi_{0}\right) g_{uv} + \left [ \sum_{k} n_{k}w_{k} \right ] g_{uv}[/tex]
The first term on the right is the classical part of the cosmological constant, the latter term on the right is the first oscillator correction to the scalar field's classical value. The sum of these two numbers + the contribution from anything you want on the left hand side (classical value, as well as quantum corrections) must be equal to the effective cosmological constant that is observed in nature. In practise people just group everything that is naively classical to one part, and everything that is quantum to the other part.
A few notes: The quantum part of this diverges as k -- > infinity, so it must be regulated. If you are completely naive and perform a hard cutoff at some length scale L, you will see that you will get an answer that depends on the fourth power of this length scale. This is part of the problem. Why should this number (presumably derived from the microphysics of fundamental physics) match a classical part to an incredible 10^120 decimal places of accuracy (the classical value is in principle sensitive to cosmological contributions, like domain walls and things of that nature). This incredible conspiracy of hierarchies is why physicists believe there is a problem with the logic of this setup, but it is very difficult to sort of undo this, without violating some other known part of physics. For instance if you try to force the quantum part to take the value zero, you also spoil things like inflation.