These hand and finger positions have a great deal of meaning. Known as mudras (Sanskrit word meaning seal, mark or gesture), they are used during meditation to direct energy flow into the body.
“According to yoga, different areas of the hand stimulate specific areas of the brain. By applying light pressure to these areas of the hand, you will “activate” the corresponding areas of the brain,” explains Sanskrit Magazine.
Mudras are described as having the same meaning to Hindu and Buddhists as folded hands and other hand gestures have to Christians.
The magazine goes on to describe some of the more common mudras:
Gyan Mudra – where the index finger and thumb are lightly touching. Known as the “Seal of Knowledge” this mudra is believed to encourage wisdom and expansion. When used during meditation or pranayama (breath work) it allegedly eases drowsiness, inspires creativity and increases concentration.
Shuni Mudra – where the middle finger and thumb are lightly touching. This is known as the “Seal of Patience”, it symbolizes patience, discernment, focus and discipline. It allegedly encourages patience, the courage to fulfill one’s responsibilities, helps one to have noble thoughts, and turns negative emotions into positive ones.
Surya Ravi or Prithvi Mudra – where the ring finger and thumb are lightly touching. Known as the “Seal of the Sun,” holding the hands in this position during meditation or pranayama allegedly causes energy, positivity, intuition and helps to encourage positive life changes. It is also believed to build and invigorate body tissue.
Buddi or Varun Mudra – where the little finger and thumb are lightly touching. Referred to as the “Seal of Mental Clarity” this hand position is believed to encourage clear and effective communication and to “help balance the water element” in the body by activating salivary glands and moistening the skin and eyes.
There are a variety of other mudras which can be found here.
I could find no scientific evidence to support claims that specific finger positions are able to produce any of the effects described above.