Why do these standing waves appear to be traveling? Take a look at this video:
https://www.youtube.com/watch?v=TKF6nFzpHBU
You would expect guitar strings to produce standing waves, and in fact, the upper, thickest string does often produce long, standing waves. Nevertheless it can be seen many times in the video that the waves are actually traveling in one direction. Is this an illusion related to the framerate of the camera, or is this effect real and caused by something?
 A: See the last animation in http://www.acs.psu.edu/drussell/Demos/SWR/SWR.html

A standing wave can appear to move if the ends exhibit any damping properties. In this case the fingers touching the strings, or the wood in the guitar (via the frets) provide the damping. This is confirmed if you notice that there are no perfect nodes in the string (points with no motion). Just areas if high and low amplitude, the ratio of which is called Standing Wave Ratio (SWR).
A: 
Is this an illusion related to the framerate of the camera

Yes, as is explained in words and video here:

This story begins not with how guitar strings actually vibrate, but a
  curious phenomenon when combining the regular oscillation of the
  string with the rolling shutter of a CMOS digital camera — namely, the
  iPhone’s. To accurately capture motion, you need to record an image
  all at once (or at least come close). Rolling shutter or line scan as
  exhibited in a CMOS camera sensor like the iPhone is a side effect of
  the capture being scanned from top to bottom, so the bottom portion of
  the image is saved later than the top. That causes motion to skew
  across the image. (Long before digital, people played around with the
  same effect in analog video and even using film photography – all you
  need is something moving and a way of capturing the image that moves
  gradually in a different direction.)
When the regular oscillation of the scanning combines with the
  oscillation of what you’re filming – as with a vibrating guitar
  string, or the rotating propeller on an airplane — the two frequencies
  effectively phase, causing some curious distortion. In the case of the
  guitar, this means seeing the appearance of standing waves that, while
  they can occur in nature, don’t occur on any conventional guitar. (You
  can also think of the basic effect as aliasing, as seen optically when
  video shutters capture the frequency of rotation of a rotating car
  wheel in such a way that it appears to move backwards.)

A: If the strings were driven by a sine oscillator at the proper frequency, the standing wave nature would appear with fixed nodes. The frame rate wouldn't affect where the nodes are. You would need the antinodes flexing up and down, but the lateral positions wouldn't change.
However, in this video several things appear to be happening. First, the pluck generates several harmonics on the string. Few of the nodes of these coincide, and as the numerous standing waves damp out for one particular note, they disappear at different rates. Also, in real strings, the overtones are not perfect harmonics causing any alignment of nodes that might ideally be present to be a tiny bit off. The nodal mismatch and the different damping rates of the overtones change the wave form on the string dramatically as a function of time, and the "strongest"  nodal positions change as a function of time. This, combined with the frame rate, gives the effect that you see.
A second-order, very small effect that might be visible (I'm not sure, but it would be a tiny effect) is that when a string is plucked, the tension is higher than normal. As the amplitude decreases, the tension decreases (remember, these are real string and have real stiffness), so the pitches change a miniscule amount. To see this, use a good tuner and pluck guitar strings hard vs. soft.
Another effect that's visible is when the player changes notes. This causes a dramatic waveform shift (especially for sliding or bending notes) which, again, is caught by the frame rate.
