What, if any, is the relationship between gravitons and gravitational waves? I understand that waves have been detected indirectly (although I’m not quite sure what that means) and that there is a near infinite likelihood that gravitons will never be.
The same as the relationship between photons and electromagnetic waves, and not really possible to explain to a five year old, or a person at any age, really. 
Here’s my understanding, as a happy amateur.
Single photons can be quite high energy and easy to produce and detect singly. Low energy photons are much more difficult to produce and detect singly, but we have no trouble producing and detecting corresponding electromagnetic waves at quite low frequencies.
Detecting gravity waves is much harder than detecting electromagnetic waves, in part because they are everywhere and there is no way to shield for them, like we can for EM. But we’ve managed.
You can’t just put up a phosphorescent screen and catch the corresponding gravity waves though, and this is from the most powerful gravity wave sources we know. And we certainly can’t create a similar source at will.
About gravitational waves, the deal is that distance or length isn’t set in some rigid linear invisible framework. It can grow and shrink. The distance between two stationary points can vary a little bit, and it depends on how much mass there is, especially nearby but even at a distance. If something changes about that mass, the change doesn’t take effect instantly everywhere. It takes time to propagate. Gravity waves are the fluctuations in the effect of mass, traveling outward from wherever the mass in question is. Big black holes, and to a lesser extent neutron stars, are so massive and can get so close to each other while rapidly orbiting that their gravitational waves are especially detectable.
About gravitons, well, I don’t know in particular – but “wave” and “particle” are two simplified idealizations that represent the same underlying thing in different ways. For light, the two different idealizations can be in pretty stark contrast to each other, and dealing with light when you have to mentally shift between the wave and particle idealizations can feel pretty paradoxical. It’s pretty easy to prove that light can’t be a particle, and has to be a wave. It is even easier to prove it can’t be a wave, and has to be a particle. Actually, it isn’t either, and a more accurate idealization that deals with the range of light behavior is just too mentally unwieldy for us mere mortals to handle (well, most of us).
AFAIK, gravity conceived in particle terms isn’t a very useful idealization, not nearly as useful as light. That’s all.
The idea of gravitons is fairly simple, but a bit above what can be explained to a 5 year-old.
General relativity describes gravity in terms of the curvature of spacetime. In particular it describes the geometry of spacetime in terms of a field on spacetime which will call M. However an alternative way is to to choose a background geometry M’ and describe gravity in terms of a physical field on spacetime λ such that M = M’ + λ. This is the ideal way to describe gravitational waves as their wavelike nature is manifest in λ.
Now that you have gravity describe by a physical field λ you can use standard techniques to quantitize λ to get a quantum theory of gravity which instantly gives you the basic properties of gravitons, which will appear when λ describe gravitational waves.
However there are problems with this approach. Classically the problems are more philosophical than practical as it is against the spirit of general relativity to separate gravity in to a physical field and the choice of M’, and hence λ, is always somewhat arbitrary. That said, classically it remains a useful tool. Using this approach to create a quantum field theory brings an additional practical problem in that the resulting theory is non-renormalizable which severely limits its ability to make physical predictions. Most would agree the few physical predictions, such as the basic properties of gravitons, it does make should be the same that a more practically-useful approach to quantum gravity would make, but otherwise the approach is seen largely as a dead-end.
IANAP but I don’t think this is correct. Gravitational waves are produced only by moving masses. A static mass like a single isolated star does not produce any gravitational waves, but it still has a gravitational field and would exchange gravitons with any nearby objects.
And a static electrical charge doesn’t produce electromagnetic waves, but still has an electric field and would exchange photons with other electromagnetic charges.
Detecting gravity waves requires measurement of a fluctuation in gravity, and some means of identifying the source of the fluctuation. The simultaneous measurement of distances in two directions among three points for which recorded distances before, during, and after a specific event have been recorded with reliable time indexes, and a precise location of the two very large objects which have moved in reliably predicted paths at a known location relative to the detector can measure the change in the distance do to warping of space by gravity. Putting all that together in real time is quite an effort. Making sure nothing else happened at the same time that shook say, the planet, or simply a continent is a non trivial aspect as well.
Five year old version. If you shake something a whole lot bigger than the whole world and you watch really closely you can see the part of the world where you are stretch, and then smoosh back together. You have to do a lot of arithmetic to figure out when it happened, and where and everything is.
Tris
naita got it right in the first reply, but I’ll just back her up: In the same way that an electromagnetic wave can be considered as a stream of photons, a gravitational wave can be considered as a stream of gravitons. But while it’s easy to detect individual photons (at least, of sufficiently high energies, but photons of sufficient energy are not rare), there are a variety of reasons why it’s extremely difficult to detect individual gravitons (as in, I believe that we will probably literally never be able to do it, and that’s not something I say lightly).
I will also add that the detection of gravitational waves is no longer just indirect. We’ve had indirect observations of them since the 1970s, when we noticed that the binary pulsar was losing orbital energy at just exactly the rate we’d expect from gravitational waves. But in the past few years, after the culmination of a massive and very difficult project, we’ve been detecting them directly.
This 7-minute video may do a better job of explaining gravitational waves and their detection, but this 9-minute video about the “absurdity” of detecting those waves may be more fun.