Maha means great. Eshwari means Lord. Maheshwari means she is the greatest of all the Lords. Mangala chandi vidhya starts from here.
Background - Vajranga is the son of Kashyapa and Dithi. Varaangi is his wife. She used to fight make devatas life miserable as much as possible. Tarakaasura, son of Vajranga meditated upon Brahma and got a boon such that there is none/nothing in this whole creation that is equal/greater than him. Then he waged war on Devatas. In the end he died in the hands of Lord kumaraswami.
Tarakaasura had 3 sons. They are 1.Vidhyunmaali, 2.Taarakaakshudu, 3.Kamalaakshudu. These three meditated upon Lord Brahma again and got 3 boons. Brahma gave them 3 cities that can travel faster than light. He also said that you three will be victorious as long as these three cities doesn't come in line to each other
Vidhyunmaali, Taarakaaksha and Kamalaaksha (Tripurasuras) called Mayudu and asked them to build empires with Gold, Silver and Iron in each of these three cities. Pura means a city. Tri-pura means three cities. These are the 3 cities Tripura represents.
Hari is son of Taarakaaksha. He meditated upon Lord Vishnu and got a boon such that if anyone from the army of these three asuras die, a 100 more will immediately emerge when the dead body is drowned in water. Then he built numerous wells in these three cities to facilitate that.
Now these asuras became invincible. With ability to travel faster than light, they used to move and land wherever they wish. They used to cause immense destruction to the abodes of devatas, Rishis and Saadhus just by landing their cities. Nobody is powerful enough to defeat/kill them. Hence Devatas reached Lord Shiva for help.
Shiva made earth as chariot, Sun and Moon as wheels, Brahma as rider, 4 vedas as horses, Meru as bow and Sri Hari as arrow and started war against Tripurasuras. In the war, Asuras destroyed Lord Shiva's chariot wheels. Then with advice from Lord Vishnu, Shiva sought help from goddess Chandi. With her help, Shiva killed Tripurasuras.
Because Lord Shiva became victorious with her help she is called Managa chandi. Teenage girls worship her on Tuesdays. Mangala, a descendant of Manu dynasty worshipped her and got mukthi.