The fainter the star, the further away it is? If a star appears to be very faint, does that imply that it's really far away, or that its intrinsic luminosity is small?
More precisely:  If I look up into the cosmos and see a star that is very faint, which is more likely to be true: (1) its intrinsic luminosity is low, or (2) the star is far away?
 A: There are several ways of answering this question.


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*We may have additional information (not including a parallax!). For instance, if we know the surface temperature of the star and its gravity, both of which can be estimated directly from spectroscopy, then the type of star leads to a direct prediction of its absolute luminosity. This in turn leads to a distance estimate.

*Stars have a distribution of tangential proper motions on the sky. In general, the closer the star, the more likely it is to have a large proper motion. Often we can calculate the "reduced proper motion plot", usually defined as the apparent magnitude plus 5 times the (base 10) log of the proper motion in arcseconds per year. A plot of reduced proper motion versus the colour of a star gives you a strong indication of whether it is a giant or a dwarf and this classification can then be used to estimate the absolute luminosity and hence distance.

*If you know nothing else, then you can construct a plot of luminosity versus distance for star of known distance and brightness. A star of a given brightness will have a distribution of possible distances that can be used as a probabilistic estimator of the true distance. I attach a diagram showing how well (or not) this is likely to work. This uses the Hipparcos revised parallax catalogue. I plot distance (in parsecs) versus apparent visual magnitude. The spread in distance for a given magnitude is very large. There is only a very weak correlation between apparent brightness and actual distance.
I also colour code the points according to their B-V colour. There are distinct bands. i.e. If we know the colour of a star we can do much better, but there are also ambiguities introduced if you do not know whether a star is a giant or dwarf. Hence, there are two bands for stars with B-V of about 1.2-1.3 - a (closer) dwarf sequence and a more distant giant sequence.

If we ignore extinction and assume sources are distributed uniformly through space then the number $N$ of sources with a given intrinsic luminosity that are found with flux $S$ varies as $N(S) \propto S^{-5/2}$. Because this is true at any luminosity it is also true for any spatially uniform distribution of intrinsic luminosities. The relevance to this question is that for stars of any type you find many more apparently faint examples than bright ones and the reason for this is that they are included from a much greater volume of space at larger distances. Under these restrictive assumptions it is always more likely that a faint object is found near to the limiting distance, $d_{max}$, of its detectability - to be exact, the average distance will be $d_{max}/2^{1/3}$.
The question however is more subtle than this and is affected by extinction, spatial anisotropy and the relative densities of different kinds of star. The probability that a faint star is close or far away very much depends on which direction you are looking. For example, if we are looking out of the Galactic plane, then a faint star is more likely to be nearby, because there are few stars at great distances in that direction. On the other hand, if you look towards the Galactic plane, you might also be able to say that a faint star is unlikely to be very distant, but this time, it is because extinction by dust limits the horizons of your observation. On the other hand, if you look in the infrared, then the star is much more likely to be distant, because you can see (through dust) much further and sample a larger volume at greater distance.
EDIT: 13/02/15
Just to give chapter and verse on this. Since the question appears to be asking about the faintest visible (naked eye) stars, I selected all objects with $5.5<V<6.5$ in the revised Hipparcos catalogue. The first plot below shows their (normalised) distance distribution. It is reasonably symmetric in log distance. The median distance of a faint naked eye star is 440 light years (i.e half are further way than this). Thus your best estimate of distance in the absence of any other information would be about 440 light years, but with a factor of $\sim 2.5$ as a 1-sigma uncertainty. Thus a "faint star" by this definition is most likely to have an absolute magnitude of about zero, and therefore be $\sim 100$ times brighter than the Sun. However, plotting the stars at around the most probable distance on a HR diagram (see next plot) we see they are not the most intrinsically luminous stars. They are mostly red giants and also some main sequence stars a few times more massive than the Sun. This is because the brightest stars are rare, so even though they can be seen in a larger volume of space, there are still not many of these rare objects in that space.
Finally, I reinforce the point I made about Galactic latitude. The bottom plot compares the (normalised) distance distributions for stars with Galactic latitude less than $\pm 15$ degrees (low Galactic latitude), with those that are more than 45 degrees out of the Galactic plane (high Galactic latitude). This plot perfectly shows the point I made about direction-dependence. Looking out of the Galactic plane we see the normalised probability distribution peaks at a smaller distance and cuts off sharply at 2000 light years as the Galactic disk "peters out".



A: A faint star can either be far away or be faint to begin with, and without more data to go on there is simply no way to tell. 
This is a big problem in astronomy, and measuring distances is one of the main challenges in understanding any given system; for more details look up the cosmic distance ladder. If it's a star you can see with the naked eye, chances are that its distance from Earth can be measured relatively simply using the parallax method, which means that we know how far away it is, and you can simply look up this distance and from it work out how intrinsically bright it is. Without this external data, though, it's not an easy thing.
For some stars, though, you can get some idea by looking at the colour. This is because the colour of the star is in general related to its instrinsic brightness. This was discovered by Hertzsprung and Russell, who measured the intrinsic brightness of stars and plotted it against their colour (or more specifically, their temperature). The result is a big diagonal streak with a large population:

Image source
This diagonal streak is known as the Main Sequence, and if you know that a star is in it then you know that the bluer it is, the higher its internal brightness. Thus if you see two Main Sequence stars of the same brightness then the bluer one will be the more distant one.
The problem is, however, that you can't know for sure that a star is in the main sequence. A reddish star, for example, could be a small, cool Main Sequence star, or it could be a bloated red giant with a much bigger surface area and therefore a much higher total luminosity. (Because of the Stefan-Boltzmann law, the brightness of a patch of star surface of a constant area is only a function of its temperature). Similarly, a bluish star can be a massive, young main-sequencer, or it can be a smallish star that has run out of fuel and shrunk down into a white dwarf. Without having more information, it's simply not possible to tell for sure.

If you're OK with a probabilistic sort of statement, though, then there are indeed more things you can say. If you look at the HR diagram above, you can see that the different populations can have radically different numbers of stars in them; for example, there's generally rather few supergiant stars. There are a number of things which affect the numbers of different stars that we can see: 


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*volume effects (where a brighter star is visible over longer distances, which means that more of the stars we see tend to be intrinsically brighter),

*intrinsic density effects (i.e. the overall probability of a star to be of a given type, so for instance in the main sequence brighter stars are less common),

*local density effects (as the local environment can differ from galaxy-wide properties), and

*absorption effects (where stars in the galactic plane are partially obscured by dust, appearing fainter than they otherwise would be)
among others. However, if you're looking at a given star and you want to be able to say things about it, then we can simply compile the statistics for the stars we can actually see, at the brightness we see them at, and then use those to try and see how much you can say about your star. 
The diagram below, adapted from Rob Jeffries' answer, shows the distance (in light years, on a logarithmic scale) and the visual magnitude of the stars that are visible to the naked eye (so bright stars are to the left and faint ones to the right). The colour of each star indicates its B-V index, and it is a rough indication of the visual colour of the star. (See the code used to produce it in the revision history.)

There's a few things to notice here. The first is that there is a pretty sizeable spread in the distances, with most visible stars between 20 and 2,000 light years. Any further than that, and they'll generally be too faint for us to see; any closer, and there simply isn't much volume between here and 20 ly away to fit many stars.
Additionally, there is definitely structure in the way the different colours occupy the diagram. Unfortunately, there's relatively little you can actually say with it, because the different star colours are still pretty mixed. Over a larger population, as depicted by Rob and with the kind of stars you might see in a telescope, there are much clearer bands, but if you restrict yourself to naked-eye objects there's rather less you can say. Nevertheless, a faint star is somewhat more likely to be intrinsically bright and far away if it is very red or very blue, whereas a whiter star is somewhat more likely to be not-so-bright intrinsically, and somewhat closer by. To the small extent to which you can say anything, though!
A: You need some standard candles and spectral analysis or you have to measure the parallax, otherwise you can't tell for sure. If the object is far enough away that its velocity due to the expansion of the universe is bigger than the peculiar velocity you get a good estimate for the distance by measuring its redshift.
A: Chances are it's really far away. Whilst stars are fairly evenly spread over the faint-to-bright scale, simple geometry dictates that the vast majority of stars are very, very far away, so for any random faint star (with no other information available) the chances are it's down to distance rather than being at the particularly faint end of the range.
This is all assuming we're using a big telescope. If your question specifically refers to naked eye observations, then most of the stars you can see in the sky are very close, within a few thousand lightyears, and most of those are very luminous, even the "faint" ones, so for naked eye observations distance is also the primary factor.
