What makes a Feynman diagram real or virtual? Simple question: as the title says, what makes a real Feynman diagram real, and what makes a virtual diagram virtual? Or in other words, how do I tell whether any given diagram is real or virtual? I've never gotten a really satisfying explanation of this. I would imagine it has something to do with virtual particles, but all internal propagators are virtual particles and I know for a fact that having internal lines doesn't make a diagram virtual.
 A: If I understand your question correctly its just a matter of what you are calculating whether you put the external particles on shell or not. If you are, for example, calculating an amplitude to use for a cross section, you'll put the external particles on-shell and it will be what you call a 'real Feynman diagram'. If you are calculating an effective action you might draw the same exact diagrams as you do for a cross section calculation, but you leave the particles off-shell, or what you are referring to as a 'virtual Feynman diagram'.
A: Feynman diagrams are just that: diagrams.
Real or virtual is what the particles depicted in them can be. A distinction should be made:
In order to calculate an amplitude, one needs to integrate over all possible momenta of internal lines. Therefore, those propagators can be thought as virtual. Effectively, one sums over all virtuality levels of the internal lines.
External lines are not as simple. While most are usually real, many are not. For instance, you can produce a pair of Z bosons from a total energy smaller than twice the Z boson mass. In that case one of the Z bosons will be virtual. If you measure it, you'll not find the Z boson mass, but something lower.
The same applies to the Drell-Yan process, where a virtual photon is created from quark-antiquark annihilation. These photons are massive, while real photons are massless.
Bottom line: diagrams are what they are. What comes out of them can be on-mass-shell (real particle) or off-mass-shell (virtual particle).
A: In the normal usage, real and virtual are not properties of Feynman diagrams themselves, but of the particles depicted in them. The particles corresponding to external lines (attached to at most one vertex only) are real, the others (attached to two vertices) are virtual.
A Feynman diagram may be considered as a repetitive part of a bigger diagram. This requires that the external lines of the diagram are kept off-shell, so that the corresponding integrals depend on off-shell momenta (rather than only on-shell momenta, which would suffice for S-matrix elements). Their computation is more complex as one has to account for a bigger parameter space. The use of off-shell Feynman diagrams in this sense is that they can be used in resummation techniques as building blocks of infinite families of on-shell Feynman diagrams. Indeed, such a recursive usage necessitates treating the external lines as virtual. 
As with virtual particles, such off-shell Feynman diagrams have no measurable counterpart but are just intermediate expressions in the resummation calculation.
However, in papers such as http://www.sciencedirect.com/science/article/pii/0550321379901160, a virtual Feynman graph is simply an ordinary Feynman graph involving a loop, in contrast to a real Feynman graph = lowest order tree graph. See the explanation after eqs. (42) and (81). This paper is quoted in http://arxiv.org/abs/1012.0507 after eq. (3) for details about virtual diagrams. In http://arxiv.org/abs/1112.1061, this terminology applies to the graphs cut at the lines in Fig. 1, corresponding to a factorization. 
http://arxiv.org/abs/hep-ph/9701284 also uses this terminology; see the titles of Sections 3.1 and 3.2, and the corresponding figures.
