broegger,
On a guitar string the fundamental is one half of a sinewave. So it starts at 0 at one end, goes up (or down), and then back to zero at the other end. So when the string vibrates at its fundamental frequency, every place along the string is going up and down (in synch), with the amplitude of the vibration greatest in the center.
All the overtones (or harmonics) are odd multiples ( n=3, 5, 7...) of one half a sinewave. So the first harmonic starts at 0, goes up, then down, then up, then back to zero at the other end. And so on. So when the string vibrates at say the n=3 harmonic, there are 2 nodes (no vibration) at 1/3 and 2/3 along the string, and in between is a shape (going up and down) just like the fundamental except 1/3 as long, and going 3 times as fast (which is why it's at a higher pitch).
Taken all together (1, 3, 5, 7...) these possible individual vibrations are called the string's normal modes of vibration. The string can only vibrate as some linear combination of these modes. That means the shape of the string when it's at its maximum displacement looks like:
fundamental: a1*sin(1*2*pi*x/L)
or fundamental and the n=3 harmonic: a1*sin(1*2pi*x/L) + a3*sin(3*2*pi*x/L)
or n=1, n=3 and n=5: a1*sin(1*2pi*x/L) + a3*sin(3*2pi*x/L) + a5*sin(5*pi*x/L)
and so on.
Each of those sums is a single function of x, the distance from one end of a string with length L. So the string only has one shape at a time.
By the way you can play around with this on a spread sheet by defining a function as the sum of the first five or ten modes, and then make of chart (graph) of the function. Be sure to have the a1, a3, a5... in your function be references to cells on the sheet so you can easily change their values and see what happens. For example, give this one a try: a1= 1, a3= 1/3, a5= 1/5...