There are three types of vibrato - arm, hand and finger. These definitions refer to which part of the arm or hand is the most active.
The forearm (lower arm) is active and the hand and fingers passively respond to the movement of the forearm.
The hand is active and the rest of the arm and fingers passively respond to the movement of the hand. Hand vibrato is sometimes referred to as wrist vibrato, but since the wrist like all joints is always passive, it is more accurate to call it hand vibrato. The wrist joint, like all the joints of the arm and hand (shoulder, elbow, wrist and finger joints), needs to be flexible and available to move when needed.
The finger is active and the hand and arm passively respond. Finger vibrato is primarily used to add extra colour to an arm or hand vibrato and as such is usually studied only after both the arm and hand are reliably known.
When to begin vibrato
You can learn vibrato as soon as the left hand position has settled and all fingers can lift and fall within a stable hand frame. Do not start developing vibrato too late since a functioning vibrato can help develop release in finger fall, an awareness of arm steering and a general release in the left hand. However be careful not to develop vibrato before the basic hand frame is settled since confusing the two actions can cause problems. For instance rolling the hand from the wrist is part of hand vibrato but not part of the finger fall action. If a student is rolling the hand to place the 3rd and 4th fingers then it is best to stop vibrato until the finger fall issue is resolved.
In terms of which vibrato to learn first, the wisest course seems to be to use either arm or hand to begin, whichever seems to come most naturally. When either the arm or hand vibrato is relatively settled, then you can work on the other. In the long term players will develop both arm and hand vibrato
Vibrato is a cyclical action (like the sautillé bowstroke). It starts with a downward action towards the scroll, followed by a rebound back towards the bridge. Developing the rebound is important since if we try and control each individual part of the vibrato action
it will be stilted and jerky.
Knocking the peg gives a temporary wall for the finger to hit and rebound from.
See Pre-Vibrato Knock the Peg
You need a flexible nailjoint to allow the elongation of the finger to slightly flatten the pitch then as the finger recurves it returns to the correct pitch to complete the vibrato cycle.
The thumb plays a balancing role (like the upper arm). Take care when learning vibrato that the thumb is stable when you change from a wide shifting-style movement back and forward along the string to a narrower action and then settle on the note.
The release of double contact
Swing the upper arm slightly to release the double contact between index finger and the neck. This slightly changes the hand position so that the fingers are flatter.
The direction of the vibrato action is important. Since our left-hand position is always somewhat sloped, our fingers contact the string at an angle. This in turn means that when vibrating (with either arm or hand) we need to use the 'give' in the finger joints to allow our pliable finger pads to maintain contact with the string. Keeping the finger pads soft and pliable allows our fingers to passively respond to the vibrato action with a slight rotary movement.
Passive balancing movement
A passive balancing movement in the upper arm is important. Check that your violin set up properly supports the violin since if the left side is too tight the vibrato action will be compromised. Paul Rolland and Sheila Nelson both describe the importance of the passive balancing movement in the upper arm.
Practise with and without vibrato
It's good to practise without vibrato so that we can monitor the warmth and resonance of the sound when we add vibrato. It is also useful to monitor how the bowing action affects the vibrato.
Finger pressure and width and speed of vibrato
Move between light harmonic touch to quite firm pressure of the finger to the fingerboard. See Developing Flexibility
Pre-Vibrato - Knock the Peg