A Yagya is a spiritual sacrifice.
In a common use of the word Yagya means - a fire ceremony commitment and performance of worthy cause..
It is performed like this - a Vedee (an altar) is built, sacred fire is set in it, and Aahuti (oblations) are offered in it along with Mantra. Otherwise there are many kinds of Yagya. All the names of Yagya and their description is not possible here, but some information about them is given here.
Yagya has another common meaning that is, when one sacrifices anything for the welfare of others, such is Bhoomi Yagya, Gyaan Yagya, Shikshaa Yagya etc it is called Yagya. Even our life is a Yagya, if we wish so, we can spend it for the cause of others.

Two odd kinds of Yagya are well-known in Hindu scriptures -
(1) Janamejaya's Sarp Yagya (Snake sacrifice). which he did to take revenge from Takshak Naag because he bit his father Pareekshit. According to MBH, 0/9 his Yagya was stopped by Aasteek, the son of Vaasuki Naag's sister Jaratkaaru.
(2) Vashishth's grandson Paraashar's Raakshas Yagya which he did to take revenge from Raakshas community because one Raakshas killed his father Shakti. His Yagya was stopped by Pulastya Muni.

Bhavishya Puraan, p 246, says - Whoever is without son and wife, has no right to do Yagya.
Mainly there are 21 types of Vaidik Yagya,
(1) Sapt Paak Yagya are 7;
(2) Sapt Havir Yagya are 7;
(3) Sapt Som Yagya are 7.

Parts of a Yagya
Setting Up -
In all kinds of Yagya, the Divinity for whom the Yagya is doene, is invoked into a ritually purified and sanctified fire altar called a “Hom Kund.” God is praised with powerful Mantra. Prayers are offered to the Lord as ghee and special healing herbs are offered into the fire. The subtle essence of these offerings is restructured by Gods and Devtaa. The fragrant smoke becomes imbued with miraculous healing properties that enrich and renew all they touch. Vaidik Yagya like the Aayush Yagya (done specifically for healing and long life) have a unique ability to transform / sanctify the people who participate and the places where they are performed. The Yagya is the most complex and powerful ritual the Vaidik tradition has to uplift peoples consciousness. Such rituals actually install the Divine qualities (in this case the Divine capacity of healing) into people and physical matter. The effects of such rituals are extremely powerful to alter the history of whole geographical regions. While medicines work on the physical body, Yagya heals the body, mind and soul. It actually awakens the Divine capacity of healing within the body at all levels. All diseases are destroyed by the faithful participation and this sacred offering. Yagya is also called Hom and Havan.
Sankalp –
Here, the nature of the Yagya being preformed is announced and described in Sanskrit. Sanskrit is a mystical language that is structured in a way that the Mahadevas and devas can easily hear it (even in places where the dense worldly vibrations usually make it difficult for the angels to see and hear people). Sanskrit is a divinely revealed language. Its nature is to inspire human consciousness toward divinity. To hear the sacred sounds of Sanskrit is enough to transform human consciousness. The Sankalp is preformed as an announcement for the Devtaa, and an affirmation for the worshipper. Each person present will be offered an opportunity to state their name and any specific prayers they may have at this time. Once the Sankalp is made, the ritual described must be successfully completed.
Ganapati Poojaa –
Ganapati Poojaa is the worship of the beloved Mahaa Dev (Archangel) Ganesh. Though there is only one formless Absolute, there are many Devta who act as His representatives and servants. We as humans, come to know that Formless One through their loving guidance. While God is manifest throughout the universe, the Devtaa embody specific (limited) Divine qualities. They each have unique abilities to help us with different obstacles we may encounter in our spiritual practice. Ganapati (Ganesh) is always worshipped first to remove obstacles the work is in habd. His worship (Poojaa) gives us the wisdom that leads to attainment and success.
Yagya Shaalaa Punya Vachan –
This is the ritual purification of the Yagya Shaalaa (the place of the fire sacrificee). It is done by blessing water in special Kalash (water pots) with Mantra and then sprinkling that water around the place. Also the offerings are made for the Guardians of the 10 directions and for ghosts and malefic spirits that might otherwise seek to harm the effects of the ritual.
Yagya Shaalaa Pratishthaa –
Yagya Shaalaa Pratishthaa is the consecration of the place by rituals of invocation.
Yantra Pratishthaa –
Yantra is a mystical diagram used in worship and meditation. It is the visual form of the Mantra and the Deity. It is like the Tibetan Mandal. A special Yantra and Mantra has been revealed to the Swaamee for use in healing rituals, for example in Aayush Yagya, it is the Yantra / Mantra of Baal Vaidyanaath (the divine strength of healing) is used. The Yantra has been carefully constructed with colored cream of wheat. Yantra Pratishthaa is the ritual consecration of the diagram. Divine Shakti (Power) is installed into the Yantra by various Mantra used to energize the various parts of the diagram. After this ritual the Yantra is respected as a manifestation of God Himself until it is deconsecrated. The consecrated Yantra is extremely powerful to bless the people who have its sight).
Yantra Poojaa –
Yantra Poojaa is the worship of the God of the Yantra by Shodashopachaar (the 16 traditional offerings).
Kalash Sthaapanaa –
Kalash Sthaapanaa is the invocation of God’s Shakti (power) into special water pots (Kalash). These are simple an-iconic images of God. The water is blessed by the process to add power and auspiciousness to the Yagya. There will also be Kalash filled with Akshat (raw whole colored rice). These are sacred to Lakshmee Jee, the Goddess of wealth. This ritual will be abbreviated to make time for other events.
Agni Pratishthaa –
The fire is a symbol of the Divine power of transformation, transmutation, digestion and consumption. The fire is the spiritual fire of awareness burning at the third eye, and it is the fire of spiritual austerity that burns away our faults and impurities. Divine consciousness pervades all the universe. The Ved teach us to literally worship God in natural things throughout the universe as a means bringing consciousness into union with the Divine. God is invoked into the fire after rituals of purification and consecration of the Agni Kund (the fire altar). God in the fire, consumes the offerings , carrying their subtle essences to the inner worlds where the offerings are received by the Devtaa. These offerings are restructured by the angels and used to fill the atmosphere of the Yagya Shaalaa with Divine Shakti (Power or Energy). In the Agni Pratishthaa (this means literally enlivening of the fire; or awakening of the Divine in the fire) Lord Agni (God of Fire  or Transformation) is invoked and worshipped.
Ganapati Hom –
After Agni Ptatishthaa, the next, offerings are made to the fire for Lord Ganesh.
LakShmee Hom –
Then Lakshmee the Goddess of Wealth and Prosperity, Hom is performed. Her Hom ensures a prosperous Yagya.
Rudra Hom –
After Lakshmee Hom, Rudra Hom is performed. It is the Vaidik form of Lord Shiv (God of Peace and Transcendental consciousness). His Hom destroys our obstacles and faults, binding us with the highest Bliss. His Hom is praised by the scriptures as the highest form of worship. There is no attainment that cannot be had by the proper worship of Lord Shiv.
Aayushya Hom –
If one is doing Aayush Hom, then Aayush Hom is done. This is the Vaidik Prayer for Health. It is an ancient and powerful rite.
Dhanvantari Hom –
Dhanvantari is the Name of the Lord of Healing. The Lord’s healing nectar empowers every act of healing throughout the world. This offering helps bring the Divine power of healing into those who chant or hear the Mantra. For this ritual emphasis will be on Jap (repetition) of His Mool Mantra (Root Mantra). Also, the Bala Vaidyanaath Moolaa (this powerful Mantra has been revealed to the Swaamee Jee and it relates to the Yantra used in this ceremony). The Mantra will be repeated. People are encouraged to chant along and make offerings to the fire.
Mrityunjaya Hom –
The Mrityunjaya Mantra is the Mantra of Lord Shiv as the conqueror of Death. For this Hom, special Beej Mantra (sacred syllables) enact the healing properties of this Mantra. This Mantra is very powerful practice for all who seek Moksh (liberation) from any form of suffering or worldly bondage.
Bhaishajya Guru Hom –
Bhaishajya Guru is the Sanskrit name of the medicine Buddha. Because we are at the sight of a weekly Buddhist Satsang where the Tibetan Moola Mantra of the Medicine Buddha is chanted, we will make offerings with this sacred Buddhist Mantra. The word “Soha” at the end is equivalent to the Sanskrit “Swaahaa”. It means “I offer this” and also “I am one with God.” It is used specifically for making offerings to the sacred fire.
Dravya Aahuti –
Dravya Aahuti is the offering of herbs and other substances to the sacred fire. Hundreds of powerful Aayurvedic and Chinese herbs have been specially prepared for this ritual. Also many local herbs have been selected and gathered to help connect this ritual to the natural surroundings. The Swaamee has used his extensive knowledge of Herbs and Aayurved to prepare thousands of dollars worth of medical quality botanicals. The Devtaa have guided him at every step, so that he may choose only those herbs best for this ritual and the people participating. Even though the ritual has not been preformed yet, there have been reports of miraculous healing from those participating (including a strange “incurable” bone condition in the hand of a participant).
Poorn Aahuti –
The Poorn Aahuti is the final offering made in all Yagya. It is made after all the other offerings have been successfully completed. The Swami will not make the purnahuti until the other offerings have been made to the satisfaction of the Devtaa. This ensures that the best results possible are achieved. Its effects are powerful, and that he does not have any hesitation to continue throughout the night or even for several days, if he feels it is necessary. He expects that the Yagya will be fulfilled by early evening, but will listen to the inner guidance of His Guru and Devtaa to make any useful adjustments to this schedule. The Poorn Aahuti is the pinnacle of the Hom. It is very auspicious to be present when this offering is made. It is said to bestow the effects of being present for the duration of the Yagya. The only way to ensure best personal gain, though is to participate as completely as possible. Only when the Saadhak completely surrenders the ego and offers all the desires of the separate self into the sacred fire will the highest benefit be had. That Saadhak verily becomes the God he worships.
Vasor Dhaaraa –
“The stream of excellence” ritual is done for feeding the fire after (or sometimes before) the Poorn Aahuti. A stream of ghee is poured into the fire while chanting the Chamakam of the Vaidik Rudra hymn. For this Hom, sesame, coconut, and mustard oil will also be poured for the Aayurvedic constitutions (Vaat, Pitta, Kaph). This ritual increases the benefits of the Hom. Usually the Purush Sookt and other Vaidic Mantra are chanted at this time… this sacred Mantra describing how God in the ritual and God in the universe are the same, while be chanted later.
Kumbhaabhishekam –
This is a powerful ritual of empowerment for places of worship. The images of the deities are ritually bathed with the water from the Kalash after the Hom. This is the highest of Vaidik or Aagamik temple worship.
Mantra Pushpam –
Is prayer with flowers, symbolizing the flowering of spiritual consciousness.
Ved Parayanam –
This is the chanting from the Ved. Mantra from all four Ved are chanted along with special other Mantra. The Priest will chant a Mantra of healing from each of the four Ved and then he will chant the Purush Sooktam (described above) and a Mantra from the Saam Ved.
Archanaa –
Archanaa is praise of the Lord with sacred names.
Aaratee –
Aaratee is prayer with light symbolizing the Light of Divine illumination. Then blessings are received through the flames by wave the hands before the light and touching the hands to the closed eyes.
Prasaad / Aasheervaad –
Prasaad will be distributed and people attending the Yagya will have the opportunity to take the Aashiravaad (blessings) of the Priest conducting Poojaa.