Matsyasana is a reclining back-bending asana. The name is derived form the Sanskrit matsya, meaning “fish,” and asana, meaning “pose.” Matsya is also the name of an incarnation of the god, Vishnu, who manifested himself as a large fish in order to save the earth from a flood. As Matsya, Vishnu was able to carry wise Hindu sages to safety, thus preserving the wisdom of all of mankind.

In matsyasana, the body is thought to resemble the shape of a fish. It is said that practicing this posture can connect the individual to the balance of earth and water, through the grounding of the limbs into the earth, and the lifting of the chest like the crest of a wave

This asana is useful for opening the chest, stretching the intercostal muscles between the ribs, improving circulation to the breasts and lungs, stimulating the liver and kidneys, and relieving tension in the neck and shoulders.. It is part of the primary series of Ashtanga yoga and is often performed in Hatha yoga as a counterpose for sarvangasana, or shoulder stand pose.People with back or neck injuries should consult an experienced teacher about modifications using a block or folded blanket for support, or avoid this pose. Those who have migraines, or high or low blood pressure should proceed with caution

• Sit in dandasana (staff pose), legs straight out in front of you. Press your hands (or fingertips) into the floor at your sides. Broaden your collarbones and lift your chest. Point your elbows back; if your arms are long enough, bend your elbows back as you draw the bottom tips of your shoulder blades in and continue to widen your collarbones.
• Now move your hands back an inch or so. Roll your shoulders back, point your elbows back, bend your elbows, and lift your chest. Exhale from the point between the bottom tips of your shoulder blades to the base of your skull. Lift and open the back of your heart. Lengthen that upper part of your spine, and notice if any tension has begun to form in your neck or at the base of your skull. If so, release it. Keep the back of your neck long (avoid lifting your chin and shortening it).
• Now, lie on your back, knees bent. Lift your hips and bring your hands under your buttocks, palms down. Straighten your legs. (If this strains your back, keep them bent.) Roll your shoulders back, and press your forearms into the floor. Draw the shoulder-blade tips in toward your front body, and use that action to lift your chest.
• Continue to lengthen your spine from between the shoulder blades to the base of your skull. Keep lifting the chest and lengthening the back body to bring your head back until it just touches the floor. Depending on how high your chest is, either the crown or the back of your head will approach the floor. Don’t shorten or “crunch” the back of your neck in an effort to get it down. All weight should be on your forearms and elbows, not your skull! Your head should float free, with your hair just grazing the floor. Imagine it waving gently like seaweed. Relax your neck and face.
• Keep pressing your forearms down and lifting the tips of your shoulder blades up toward your heart to support the lift of your breastbone. Draw your kidneys and vertebral column in toward the front body, without gripping your back muscles or digging the rim of your sacrum into your back. Draw the tops of your thighbones down. Intend your tailbone toward your heels. Let go of all muscular action that is not directly involved with maintaining the shape of the asana: Soften your eyes, jaw, and throat. Imagine yourself floating, supported by the living water beneath you. Stay for as long as you are comfortable, then lower your chest and head to the floor, remove your hands from underneath, and roll slowly to a seated position.

Information sourced from : The Mythology Behind Matsyasana (Fish Pose) written by Zo Newell for yogapedia.com