Why are there these other "harmonics" (violin-specific terminology) in violins? I've often read explanations of how standing waves form on the string of a violin and their harmonics but there is another phenomenon I've never seen explained:
When you play certain notes, musicians call them "harmonics" and for them, you only have to touch the string to get them to "sound". They appear at very specific points along each string and if you just touch positions above and below the harmonics, you just hear a screechy noise.
Why do these harmonics form?
 A: When you play an un-stopped string, your bowing or plucking usually excites many different vibration modes (a.k.a., "standing waves"), and what the musicians call the timbre of the sound is determined by the relative amplitudes of the different modes.
Each mode has its own frequency. Musicians call the lowest frequency the fundamental, and they call all of the higher frequency modes harmonics. The harmonics of a vibrating string always are exact integer multiples of the fundamental.

Each mode has two or more so-called nodes---locations along the string where the amplitude of the standing wave for that mode is zero. 
As @alephzero alluded in a comment, if you lightly touch the string at any point, your finger damps (i.e., it sucks all the energy out of) any mode that does not have a node very close to that point, but if there are modes that do have a node where you are touching, those modes can continue to vibrate.
Every mode has nodes at the bridge and at the nut.  If you touch the string there, then you will not change the sound.
The fundamental mode does not have nodes any place else, So if you lightly touch the string any place else, you will not hear the fundamental: You will only hear harmonics.  (More specifically, you will only hear harmonics that happen to have a node at the point you are touching).
A: To add - or clarify a small point: 
When a musician plays non-harmonic ("regular") stopped notes on a string, their fingers are effectively shortening vibrating length of the string. That is, the part of the string on one side the finger vibrates, the part of the string on the other side of the finger no longer vibrates. This new, shorter string is vibrating on the fundamental.
When a string musician plays a harmonic, they are only stopping the string at the point they are touching the string, so that the lengths of string on both sides of the finger vibrate at the same time.
A clear harmonic sound is produced when the pitch of the string length in front of the finger matches the pitch of the string length behind the finger OR when the string length on one side of the finger can be equally divided into lengths that match the length on the other side of the finger.
