That one looks a bit familiar to me!
Short, simple answer
(extracted from the longer answer below...)
Measures of functional connectivity are just correlations (correlation here referring broadly to statistical relationships, not a specific statistic): they do not say anything about the causes of those correlations including whether they are direct or indirect. The "connectivity" is functional in that it is measured only by the function at the measured nodes.
Functional connectivity refers to temporal correlations in some measure of nervous system activity. Both positive and negative functional connectivity is possible. The canonical example is correlation in the BOLD fMRI signal, which is often defined as the Pearson correlation in the resting state BOLD signal time series between remote voxels, or across blocks during some task.
For EEG, functional connectivity measures include coherence, amplitude correlations (such as the Pearson correlation of the signal envelope, typically band-pass filtered into named bands like delta/theta/alpha/beta/gamma), and measures of phase relationships such as the phase lag index and weighted phase lag index.
What these measures share in common is that they record some statistical non-independence in the activity measured in different areas of the brain. However, they differ greatly in temporal and spatial scale depending on the modality and subsequent signal processing.
Importantly, measures of functional connectivity are just correlations (correlation here referring broadly to statistical relationships, not a specific statistic): they do not say anything about the causes of those correlations including whether they are direct or indirect. The "connectivity" is functional in that it is measured only by the function at the measured nodes.
For example, consider a weighted directed graph of nodes A, B, and C. Even if there is no edge connecting A and B, if C drives both A and B then there will be functional connectivity between A and B.
You would even expect to observe functional connectivity between the brains of different people if they were, for example, watching the same video (as an aside...I think someone actually did this experiment, leading to some bogus claims about universal consciousness or something like that...the real explanation is more like "when two people listen to the same audio track, in which sometimes there is speech and sometimes there is no speech, they both have more activity in their auditory cortex during speech" which is considerably less profound...).
A synaptic connection between nodes P and Q almost certainly implies there will be functional connectivity between P and Q (though it may not be measurable depending on the strength of that connection and the structure of the rest of the network), but the inverse is not true: functional connectivity is insufficient to claim a real synaptic connectivity.
Personally, I wish another term was used besides connectivity but this is the field standard terminology.
Functional connectivity can be distinguished from two other types of connectivity: effective and structural.
Effective connectivity refers to model-based measures of causal influence of some brain area on another. Examples include the use of dynamic causal modeling (see for example Friston 2009, which is also highly relevant to the rest of this answer) and multivariable autoregressive modeling (see for example Cheung et al 2010).
Structural connectivity refers to the physical axonal connections between brain areas. In neuroimaging, it is often measured using diffusion tensor imaging which takes advantage of the fact that water more freely diffuses along axon tracts rather than perpendicular to them. Structural connectivity can also be measured more directly using neuronal tracers (dyes).
The term structural connectivity is used both in the context of connection strengths between certain areas relative to other areas, as well as to overall white matter integrity which can be disrupted in disease.
Note that while structural connectivity is often expected to correlate with functional connectivity, it doesn't need to.
These other terms are mostly used in the broad areas of neuroimaging and EEG/MEG. Using these modalities, even effective and structural connectivity only loosely imply synaptic connectivity. Detecting true synaptic connectivity requires more invasive electrophysiological techniques such as paired patch clamp recordings, or via anatomical reconstructions using tracers or electron microscopy. Synaptic connectivity can also be observed in real time using live-cell imaging of dendritic spines after injection or genetic expression of a fluorescent marker.
Cheung, B. L. P., Riedner, B. A., Tononi, G., & Van Veen, B. D. (2010). Estimation of cortical connectivity from EEG using state-space models. IEEE Transactions on Biomedical engineering, 57(9), 2122-2134.
Friston, K. J. (1994). Functional and effective connectivity in neuroimaging: a synthesis. Human brain mapping, 2(1‐2), 56-78.
Friston, K. (2009). Causal modelling and brain connectivity in functional magnetic resonance imaging. PLoS biology, 7(2), e1000033.
Jones, D. K., Simmons, A., Williams, S. C., & Horsfield, M. A. (1999). Non‐invasive assessment of axonal fiber connectivity in the human brain via diffusion tensor MRI. Magnetic Resonance in Medicine: An Official Journal of the International Society for Magnetic Resonance in Medicine, 42(1), 37-41.
Nägerl, U. V., Willig, K. I., Hein, B., Hell, S. W., & Bonhoeffer, T. (2008). Live-cell imaging of dendritic spines by STED microscopy. Proceedings of the National Academy of Sciences, 105(48), 18982-18987.
Vinck, M., Oostenveld, R., Van Wingerden, M., Battaglia, F., & Pennartz, C. M. (2011). An improved index of phase-synchronization for electrophysiological data in the presence of volume-conduction, noise and sample-size bias. Neuroimage, 55(4), 1548-1565.