Harmonic Series 

Partial Number 
Ratio to Fundamental 
Scalar Ratio 
Cents from Fundamental (Adjusted) 
Note Name 

1  1/1  1/1  0  C 
2  2/1  1/1  0  C 
3  3/1  3/2  702  G + 2 
4  4/1  1/1  0  C 
5  5/1  5/4  386  E  14 
6  6/1  3/2  702  G + 2 
7  7/1  7/4  969  Bb  31 
8  8/1  1/1  0  C 
9  9/1  9/8  204  D + 4 
10  10/1  5/4  386  E  14 
11  11/1  11/8  551  Gb  49 
12  12/1  3/2  702  G + 2 
13  13/1  13/8  841  G# + 41 
14  14/1  7/4  969  Bb  31 
15  15/1  15/8  1088  B  12 
16  16/1  1/1  0  C 
17  17/1  17/16  105  C# + 5 
18  18/1  9/8  204  D + 4 
19  19/1  19/16  298  Eb  2 
20  20/1  5/4  386  E  14 
21  21/1  21/16  471  F  29 
22  22/1  11/8  551  Gb  49 
23  23/1  23/16  628  F# + 28 
24  24/1  3/2  702  G + 2 
25  25/1  25/16  773  Ab  27 
26  26/1  13/8  841  G# + 41 
27  27/1  27/16  906  A + 6 
28  28/1  7/4  969  Bb  31 
29  29/1  29/16  1030  A# + 30 
30  30/1  15/8  1088  B  12 
31  31/1  31/16  1145  B + 45 
32  32/1  1/1  0  C 
These are the first 32 partials of a theoretical string. What this means is that when you pluck a guitar string tuned to "C", you get all of these notes. The fundamental is the loudest, the first harmonic is the second loudest, and soon. The 32nd partial is pretty darned quiet and you probably can't even hear it.
These partials occur because while the string vibrates primarily as one unit, there is a slight difference in vibration between halves of the string, thirds of the string, fourths, and so on. If you touch your finger lightly to the string at exactly 1/5th its length, the fifth partial (or fourth harmonic)* will sound. This is because your finger is deadening all the vibrations of the string except where the string vibrates in 5 parts. (actually, it may vibrate a bit in 10, 15, and 20 parts as well because they are multiples of 5 and are not muted by your finger.)
Partial Number: The fundamental and all harmonics are called Partials. The lowest of these is the first Partial and they are numbered sequentially upwards. This number happens to also represent the number of equal sections of string that vibrate to make that partial sound.
Ratio to Fundamental is the ratio in hertz between the harmonic and the fundamental.
Scalar Ratio is the ratio you would get if you adjusted all the notes to appear in the same octave. You achieve this by dividing the ratio by the appropriate power of two, (equivalent to transposing down a number of octaves) so that your result is a ratio between 1/1 and 2/1.
Cents from Fundamental represents the cents value of the adjusted Scalar Ratio.
Note Name This gives the traditional note name (assuming that the fundamental is a "C") associated with the scalar interval.
* Partials and Harmonics are just about the same thing when talking about wind and string instruments. Harmonics refer to notes sounding above the fundamental as predicted by the harmonic series. So the fundamental is like Harmonic number 1 if you will. The fundamental is also considered the lowest partial or partial number 1. So the partial number is the same as the Harmonic number. Bells and percussion and certain other instruments do not exhibit the harmonic series shown here, but they do have partials. The partials are arranged more or less randomly, but are still numbered with the lowest partial being 1.