What makes a standing wave a wave? Well, this is my physics professor's question and it really made me think a lot about standing waves, realising I don't actually understand it. What makes a standing wave a wave? How could I explain it? What actually makes wave a wave?
 A: While user @John Rennie provided an excellent conceptual and mathematical answer, I would like to also include a visual representation.
The following desmos animation: https://www.desmos.com/calculator/eqwupfmfcf shows how the sum of two identical waves travelling in opposite directions will generate a standing wave.
A: This answer expands on the answer by John Rennie.
About the characteristics that makes a physicist regard some phenomenon as a wave.
A wave involves oscillation, but it's more than that. A pendulum is an example of an object that when excited will oscillate, but that doesn't amount to a wave.
Wave behavior obtains when you have a row of pendulums, and each individual pendulum transfers some of its displacement to its neighbour. What that does is that you get propagation. See for instance this demonstration by Steve Mould
The propagation can occur in many different ways, but its that aspect of propagation by way of transfering displacement to neighbouring entities that makes a physicist say: "Yeah, that's a wave."
The wave equation incorporates both time derivative and spatial derivative. By design the wave equation is such that solutions of it describe something that is propagating.

Standing wave
Mathematically a standing wave can be described as a superposition of two counterpropagating waves of equal frequency.
A string of a musical instrument is fixed at both ends, and these end points act as points of reflection. That is how the vibration of a musical instrument string can be thought of as arising from two counter-propagating waves.
Standing waves tend to arise in circumstance where reflection is assured.
As described by John Rennie, it makes sense to acknowledge a standing wave as a member of the class 'wave', even though there is no net propagation.
A: First of all, I would like to clarify what a standing wave  is not.
Let us consider the small vertical perturbation of an orizontal string. Let us assume that when the string is unperturbed its profile  coincides with the $x$ axis.
If a vertical perturbation affects the string, the profile of the  string is pictured as $y=y(t,x)$. You see that the deformation with respect to $y=0$ depends on both space $x$ and time $t$.
The equation of (small) perturbations is the famous D'Alembert equation
$$\frac{\partial^2 y}{\partial x^2} -\frac{1}{v^2}\frac{\partial^2 y}{\partial t^2} =0\:.\tag{1}$$
Solutions of this equation are generically called "waves" since they are first of all oscillations in space when time varies. These oscillations may also propagate in space (see below), but this is not the case for standing waves.
I do not want to enter into the details of the general solution of this equation. I wish only stress that it is possible to prove that every solution can be written as a combination (actually an integral) of simple solutions of the form
$$\sin\left(k(x \pm vt)\right) \quad \mbox{and}\quad \cos\left(k(x\pm vt)\right)\:, \tag{2}$$
where $k$ takes all possible values in $\mathbb R$, so that the set of the above defined waves is a continuous set.
The elementary wave
$$y = \sin(k(x-vt))$$
represents a profile with the shape of the function $y=\sin x$ moving from the left  to the right, with a speed $v$ along the orizontal axis.
The elementary wave
$$y = \sin(k(x+vt))$$
represents a profile with the shape of the function $y=\sin x$ moving from the right  to the left with a speed $v$. A similar characterisation  holds for the cos-shaped elementary waves.
As a consequence, if you focus attention on any given point $x_0$ on the real line, you always see an oscillation when $t$ varies. That is a wave. But these waves also propagate, because the whole  profiles rigidly move in space in one or the other direction with speed $v$.
This means that the elementary waves are waves but not stationary waves.
Stationary waves are instead simple waves as before when you also impose that the string has fixed endpoints.
It happens in particular for a string of a violin or a guitar.
I am saying that the equation (1) has to be accompanied with boundary conditions
$$y(t,0)=0\quad \mbox{and}\quad y(t,L)=0\:, $$
where we have supposed that the relaxed string has a total length $L$.
In this case, the general solution of (1) can again be decomposed as a sum of elementary waves, but this time, these waves are a discrete set and have a different form:
$$\sin \left( \frac{\pi n x}{L}\right)\sin \left( v\frac{\pi n t}{L}\right)\quad\mbox{and}\quad \sin \left( \frac{\pi n x}{L}\right)\cos \left( v\frac{\pi n t}{L}\right)\:, \tag{3}$$
where $n=0,1,2,\ldots$.
Also in this case, looking at a generic  given point $x_0$ you see osillations when $t$ varies, and thus we are speaking of a wave. However  there is a fundamental difference between waves (1) and waves (3).
Waves in (3) do not propagate. In fact, differently from what I pointed out for waves (1), waves (2) have also so called nodes.
They are special spatial points $x_{n,k}$ where the perturbation of the string is always (i.e, for every $t$) $y(t, x_{n,k})= 0$. There the string does not oscillate.
As a matter of fact
$$x_{n,k}= \frac{kL}{n}\:,\quad k=0,1,\ldots, n\:.$$
This fact, the presence of nodes,  defines stationary waves.
(See @user256872 ' s answer to see a nice dynamical representation of stationary waves)
It is possible to generalize this result by studying waves of  surfaces and volumes.
The presence of nodes is of fundamental relevance in music theory (it permits both to define the notion of musical note and to play it on instruments)  and in engineering (for instance to avoid disasters due to resonance phenomena).
A: The simplest definition of a wave would be a function that satisfies the wave equation. The simplest solutions are the infinite plane waves, which in one dimension are:
$$ \psi(t,x) = A \sin(\omega t - kx + \phi) $$
but because the wave equation is linear any linear combination of solutions is also a solution. A standing wave is a sum of two infinite plane waves travelling in opposite directions, for example:
$$ \psi_{sw}(t,x) = A \sin(\omega t - kx) + A \sin(\omega t + kx) $$
Since the standing wave is a sum of two solutions to the wave equation it is also a solution to the wave equation and can therefore reasonably be described as a "wave".
The problem with this approach is that by using Fourier synthesis any function can be described as a (usually infinite) sum of plane waves, and by this measure all functions could be called "waves". However common sense suggests that we use the term "wave" for those functions that can simply be described as a sum of plane waves. Different physicists will have different views of what simple means in this context, so they will have different views on what functions can/should be called waves. However the standing wave is such a simple combination of planes waves, just a sum of two plane waves, that everyone I know agrees it can be called a "wave".
A: For me, it may suffice to say that a standing wave (a) oscillates with time and (b) has something which can be reasonably called a “wavelength” (and, of course, (c) it solves the wave equation). Therefore it is a wave.
A: As John Rennie had already written a nice answer, I would just like to add something more. Standing waves can form under a variety of conditions in a medium which is finite or bounded. But they don't usually form under any circumstances. They require that energy be fed into a system at an appropriate frequency, i.e., when the driving frequency applied to a system equals its natural frequency. This condition is known as resonance.
Any system in which standing waves can form has numerous natural frequencies. The set of all possible standing waves are known as the harmonics of a system. The simplest of the harmonics is called the fundamental or first harmonic. Subsequent standing waves are called the second harmonic, third harmonic, etc. The harmonics above the fundamental are called overtones.
A: I like the general question without being specific.
In this sense, the simplest explanation to me is that a wave is a "thing" that transports energy. In a wave you can measure energy everywhere, but not at any time.
Waves then can be added to one another. A standing wave can then be seen like Solomon Slow explained it.
A: As John Rennie says, a wave is defined in physics to be a function which satisfies a wave equation. I just want to add that what makes this a wave in common-speak is that the solutions are cyclic (i.e. repeating) functions. This is of course already clear to anyone familiar with the maths in John Rennie's answer which specifies the simplest solutions as sin waves, and that all solutions are sums of sin wave solutions (given by Fourier analysis). This does of course include standing waves, just as he describes.
