Shake hands = (verb) to hold someone’s hand and move it up and down several times as a way of greeting.

• Prime Minister Narendra Modi shook hands with Kerala Chief Minister Pinarayi Vijayan when they met at Delhi to discuss the Kerala flood situation.
• Nice to meet, Tomi said , shaking my hand when I met him at Chicago.
• People shake hands when meeting with friends and guests.
• Children should be taught how to shake hands with their friends.
• He came up and shook hands with me again before parting.

Handshake =(noun) A brief greeting or parting tradition in which two people grasp one of each other’s hand.  Using right hand is considered proper etiquette.

• Handshakes are very common especially on business meetings.
• In Russia , a handshake is performed by men and rarely performed by women.
• In business, a handshake is an important tool in making the right first impression.
• He welcomed me with a warm handshake.