Bhagwat Puran

Rudraksha subsist by the tear of lord Shiva. According to the religious text Devi Bhagwat Puran there was a Demon known as Maya who was very strong and had divine energy and power. Due to these qualities he became proud and started to trouble deities and sages. No one was able to defeat this strong demon in war. He built three towns of three different metals. Towns were built by gold, silver and iron. These towns were non-destroyable and known as Tripur. After ruling these towns by him and his two brothers he was known as Tripurasur. He started crusade against gods and easily own battle against god's king Indra. He looted angles, power and treasure of god and ruled Eden. Being helpless and powerless Indra and other gods went to beg help from Bramha. Lord Bramha gave advice to them that he could not beat Tripurasur and went to beg help from Lord Vishnu. Lord Vishnu also told them that he also cannot beat Tripurasur and suggested them to beg help from god's god Shiva. All gods including Bramha Vishnu went to beg help from lord Shiva to destroy Tripurasur and his non-destroyable towns.

Bramha Vishnu and all gods started to state their grief to lord Shiva and prayed to ruin Tripurasur and his towns. Shiva as always ready to help his devotees reveal all gods that he will ruin Tripurasur. He ordered to build a special chariot to executed demon Tripurasur. Chariot was made of earth; wheel was made of sun and moon, Lord Vishnu himself became arrow of Shiva's arch and the lord Bramha drove chariot.

After the completion of the chariot Lord Shiva lead with Bramha, Vishnu, Indra and all gods to battle and destroy Tripur. He reached the place where he can destroy Tripur by one arrow. In that time Shiva took Rudra (anger) pose. When he was ready to burst the arrow all gods became so happy and start to pray. Lord Shiva execute arrow and ruin Tripur and demolish all devil.

After the demolition of Tripur lord Shiva, Bramha, Vishnu, Indra and other gods went to Himalayas to get rest. Lord Shiva closed his eyes and meditated for some time. When he opened his lotus shaped eyes tears fell from them on the earth. Those tears decent over Sitsail hill. Sometime after those tears became rudraksha trees.The fruits that grow on this tree are known as Rudraksha Beads.  

Shiva Puran

According to another religious text Shiva Mahapuran, Parvati the divine consort of Lord Shiva asked him the origin of Rudraksha. On hearing her question Shiva said that he did penance for a thousand years. Tired of keeping his eyes closed for so long, Shiva opened them. A few teardrops fell to the ground and from these tears, the Rudraksha tree was born. The seeds of the tree were distributed on the earth.

There are different conceptions regarding Rudrakshas in Shiv Puran and other Pauranic scriptures but one thing is upheld in all of these that:-

"DIVYA VARSHA SAHASRAM TU CHAKSHURUNMEELITMAYAA,
PASCHANMAAKULAKSHIBHYAH PATITA JALBINDAVAH"

SHIV UVACH : - "For one thousand divine years I kept closed my eyes then tear drops fell from my eyes."

"TATRASHRUBINDTO JAATA MAHARUDRAKSH VRISKSHKAH
MAMAAGYAYAA MAHASEN SARVESHAM HITKAMYAYAA"

Those drops of tear which fell from my eyes gave birth to the tree of Maharudraksh on my command for the benefit of all.  

Padma Puran

Padma Puranam, Chapter 57, Sri Bhagavan said, "If a man always wears a Rudraksa with fourteen mouths, permeated with God's auspicious power, he is always honored and blessed by the demigods due to the greatness of his religious merit."
The Padma Purana states that rudrakshas may be used by Hindus of every denomination. They will even benefit atheists and sinners. Rudraksha beads are not objects of worship. Rather, they are used as worshipful aids and accoutrement. Primarily, the beads are strung into strands and worn on the body.These malas are also used in the practice of japa.