"Pranayama is control of Breath". "Prana" is Breath or vital energy in the body and "ayama" means control. SoPranayama means the mechanical system of mystic yoga, through which the airs within the body are controlled, and the senses are subdued and checked from being engaged in anything other than meditation upon Lord Viṣṇu.
Five types of prana are responsible for various pranic activities in the body, they are Prana, Apana, Samana, Udana & Vyana. Their seating place is Chest, Anus, Navel, Throat and entire body respectively.The main air passing through the nose in breathing is called prāṇa. The air which passes through the rectum as evacuated bodily air is called apāna. The air which adjusts the foodstuff within the stomach and which sometimes sounds as belching is called samāna. The air which passes through the throat and the stoppage of which constitutes suffocation is called the udāna air. Aid the total air which circulates throughout the entire body is called the vyāna air.

Subtler than these five airs, there are others also. That which facilitates the opening of the eyes, mouth, etc., is called nāga air. The air which increases appetite is called kṛkara air. The air which helps contraction is called kūrma air. The air which helps relaxation by opening the mouth wide (in yawning) is called devadatta air, and the air which helps sustenance is called dhanañjaya air.  

Through the mystic yoga practice called prāṇāyāma, the yogī controls the airs within the body to maintain the body in a healthy condition. In this way, the yogī comes to the point of trance and tries to see the Supersoul within the core of his heart. Prāṇāyāma is the means to attain samādhi,trance, in order to fully absorb oneself in seeing the Supreme Lord as antaryāmī, the Supersoul within the core of the heart.

Pranayama as per Bhagavad Gita:
Chapter 4 Text 29:
apāne juhvati prāṇaṁ 
prāṇe ’pānaṁ tathāpare
prāṇāpāna-gatī ruddhvā

Still others, who are inclined to the process of breath restraint to remain in trance, practice by offering the movement of the outgoing breath into the incoming(i.e, Rechaka), and the incoming breath(Apana) into the outgoing(i.e.,Puraka), and thus at last remain in trance, stopping all breathing(Kumbaka).

Purpose of Pranayama:
The purpose of prāṇāyāma, or mystic yoga, is to stop the mind and senses from engaging in fruitive activities. The purpose of controlling the senses is to stop one’s implication in the cycle of birth and death.The aim of prāṇāyāma is not to make the body strong and fit for working hard. Prāṇāyāma does not refer to any ulterior motive. The actual aim is to strengthen the mind and senses in order to engage them in devotional service. Whatever austerity, prāṇāyāma and mystic yoga practicesPṛthu Mahārāja performed in satya yuga, were performed, for the sake of worshiping Kṛṣṇa. Thus Pṛthu Mahārāja serves as a perfect example for yogīs also. Whatever he did, he did to please the Supreme Personality of Godhead, Kṛṣṇa.  

Pranayama Benefits
Important Pranayamas
Below are the list of pranayamas which are important to practice for the yogis in day to day life:

1.    Bhastrika Pranayama
2.    Kapalabhati Pranyama
3.    Nadishodhana Pranayama
4.    Ujjayi Pranayama
5.    Bahya Pranayama
6.    Agnisaara Pranayama
7.    Shitali Pranayama
8.    Sitakari Pranayama
9.    Bhramari Pranayama