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.

Types of Yagya
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 his grandfather.

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.

Paak Yagya
Some Yagya are done with cooked food, they are called "Paak Yagya".
Manu Smriti lists four types of Paak Yagya - 

(1) Vaishwadev Hom,
(2) Bali Karm,
(3) Nitya Shraaddh - offering food to Pitar daily, and 
(4) Atithi Bhojan - feeding a guest

Panch Mahaa Yagya
In Bhavishya Puraan, 1/6, Sumantu Muni describes five types of Yagya for Grihasth people. He says - "There are five places in a Grihasth's house where a living being can be killed - mortar, flour mill, oven, broom and the place where water is kept. To be free from this killing Paap (sin), one should do five types of Mahaa-Yagya - Brahm Yagya, Pitra Yagya, Daiv Yagya, Bhoot Yagya, and Atithi (guest) Yagya.
(1) Self study and teaching is Brahm Yagya - done for Brahm3e
(2) To do Tarpan is Pitri Yagya - done for Pitri who have given us wisdom so offer them something
(3) To do Havan for Devtaa is Daiv Yagya - done for higher species, Devtaa etc
(4) Bali Vaishwadev Karm is Bhoot Yagya - to God's creation other than human beings, feed dog, cow, crow, fish, Peepal, Tulasee
(5) To felicitate guests is called Atithi Yagya. If somebody does these Yagya regularly, he doesn't get attached to sin in spite of living in Grihasth Aashram."

The same are described by Soorya Dev to his Saarathee Arun in Bhavishya Puraqn 1/21 also --
(1) Dev Yagya - doing Havan is called Dev Yagya,
(2) Bhoot Yagya - Doing Bali-vaishwadev is called Bhoot Yagya,
(3) Pitri Yagya - doing Tarpan is called Pitra Yagya,
(4) Manushya Yagya - felicitating an incoming A-Tithi is called Manushya Yagya or A-Tithi Yagya, and
(5) Brahm Yagya - studying and teaching Ved is called Brahm Yagya.

Once somebody told me the following Panch Mahaa Yagya. These Mahaa Yagya can be done by a householder easily, that is why I am writing them here -
(1) Dev Yagya : Doing Havan. A drop of pure ghee to fire (you can use your cooking gas also)
(2) Cow Yagya : Offer food to cow.
(3) Atithi Yagya : Means offering some food to guest, you can do this by offering chocolate to children, tea to any neighbor etc.
(4) Bhoot Yagya :Offer a bit of sugar to ants, you can put fractional quantity of sugar in one corner of your house or compound.
(5) Pitri Yagya : This means giving food to your Pitri, you can do this by keeping a small Chapaatee with curry what ever you cook (Daal or vegetable), keep this on terrace around noon.

Some say the Panch Mahaayagya are -  1. Devtaa, 2. Cows, 3. Atithi (unexpected guests), 4. Bhoot (beings), 5. Braahman.

Various Yagya Performed by Kings and Rishi
Although there may be many types of Yagya, but the following Yagya have been mentioned in our scriptures Their names are given here with their references.

(1) Agnihom Yagya - Meghnaad did it
(2) Agnishtom Yagya - Raam did it ( V-Raamaayan, 7/34/99)
(3) Ashwamedh Yagya - In this Yagya a horse is sent around to establish one's sovereignty, to neighboring kingdoms asking their kings either accept his sovereignty or challenge his claim. If somebody had been more powerful than the person who was doing this Yagya, he challenged the king by seizing the horse and then fighting with him; otherwise the weaker kings gave gifts and a part of their revenue to him. If the horse had come back without any challenge, the king was supposed to have completed Ashwamedh Yagya. After the horse had come back, it was killed ceremoniously as part of the victory celebrations. Many kings had performed this Yagya in ancient Bhaarat Varsh, but the main four kings who have performed this Yagya were Bharat, Sudaas, Yudhishthir, and Janamejaya. Dasharath, Raam and Meghnaad also did this Yagya.
(4) Atiraatri Yagya - King Dasharath did it to get sons. Raam did it (V-Raamaayan, 7/34/99).
(5) Baajpeya Yagya - Ashtak, the son of Vishwaamitra and Maadhavee did it.
(6) Bahu Swarn Yagya - Meghnaad did it.
(7) Brihaspati Chakra Yagya - Daksh Jee performed this Yagya in which Satee left her body through Yog.
(8) Devasatra Yagya - mentioned in MBH, G-4-Van/15
(9) Gavayaan Yagya - Mentioned in MBH, G-4-Van/15
(10) Go-medh Yagya - Meghnaad did it.
(11) Gosav Yagya - Ashtak, the son of Vishwaamitra and Maadhavee did it. Yudhishthir did it.
(12) Gyaan Yagya - Devtaa did it ( Bhaagvat Puraan, 7/7)
(13) Ishtaakrit Yagya - Rishi Mudgal did it (MBH, G-4-Van/48)
(14) Jaruthya - Raam did it. ( MBH-Stories, 6-18-Srinjaya]
(15) Maaheshwar Yagya - Meghnaad did it.
(16) Marutsom Yagya - Bharat (Shakuntalaa's son) did it to get a son. [Bhaagvat Puraan, 9/12] [Vishnu Puraan, 4/13]
(17) Mitraavarun Yagya - Vashishth Jee did it for Vaivaswat Manu to get children. [Bhaagvat Puraan, 9/1]
(18) Mlechchh Yagya - In Bhavishya Puraan, 3/4 Pradyot did it to save his father from Hell killed by Mlechchh. Naarad Jee suggested it.
(19) Naramedh Yagya - Vyaas Jee suggested this to Yudhishthir to perform it after the war. [MBH, G-7-Postwar/4]
(20) Paitaamaha Yagya - Rishi did this Yagya when Prithu was born.
(21) Pundareek Yagya - King Gaadhi did it and gave all white horses with one black ear to Braahman in Dakshinaa. Ashtak, the son of Vishwaamitra and Maadhavee did it.
(22) Paundreek Yagya -
(23) Praajaapatya Yagya - Yudhishthir did it just before leaving this Earth for Swarg. [Bhaagvat Puraan, 1/5/15]
(24) Purushmedh Yagya - Harishchandra did it after his son Rohit brought Ajeegart's middle son Shunah-shep for sacrifice. It seems that Purushmedh Yagya is the Yagya in which a man is offered as a sacrifice, as in Ashwamedh Yagya a horse is sacrificed. [Bhaagvat Puraan, 9/4]
[It seems that Naramedh and Purushmedh are the same as both Yagya need a human sacrifice.]

(25) Putreshti Yagya - King Dasharath did it to get a son. (Maanas, 1/11)
(26) Raajarshi Yagya - Yudhishthir did it when he was in exile and Duryodhan was on his Ghosh Yaatraa. [MBH, G-4-Van/45]
(27) Raajsooya Yagya - Meghnaad did it. Yudhishthir did it to ask kings to recognize a kingdom as the sovereign.
(28) Rishi Yagya - In MBH G-4-Van/17 Yudhishthir did it.
(29) Sapt Kapaal - Devtaa did it to fill the sea with water to please Shiv. [MBH, G-9-Anu/1]
(30) Sapt Tantu Yagya - In MBH, G-3-Youth/29, Dhritraashtra suggests this Yagya to Duryodhan
(31) Sarvmedh Yagya - Vyaas Jee suggested this to Yudhishthir to perform it after the war. [MBH, G-7-Postwar/4]
(32) Satra Yaag - Raajaa Nimi was doing this Yagya when he requested his Guru Vashishth to perform it, but Vashishth told him to wait as he
       had to go to Indra, but Nimi did not wait for him ad started his this Yagya. When Vashishth Jee came back, he gave Shaap to him, and in
       turn Nimi also gave Shaap to him. Both died. Then Vashishth Jee was born again from Mitra and Urvashee [Varun and Urvashee had
      Agastya as their son]. [Bhaagvat Puraan, 9/7]
(33) Sautraamani Yagya
(34) Som Yagya - Chyavan Rishi did it for King Sharyaati to get a son. [Bhaagvat Puraan, 9/2]
       Bheem did 100 Som Yagya, that is how his son's name was Sutsom. (G-3-Youth/14)
(35) Ukothya Yagya - King Dasharath did it to get sons.
(36) Vaajpeya Yagya - King Janak did it at the time of Swayamvar of Seetaa. [Padm Puraan, 5/40]  Read about it Yagya-Special
[Is it the same as Baajpeya Yagya?]
(37) Vaamdev Yagya - Raam did it ( V-Raamaayan, 7/34/99)
(38) Vaishnav Yagya - Meghnaad did it. Duryodhan did it (MBH, G-4-V/47)
(39) Vishwajeet Yagya - Yayaati's son Puru did this [MBH, G-1-Beginning/2]; Daitya Raaj Bali did it [Bhaagvat Puraan, 8/6]

Pratyangiraa and Sudarshan Hom (not counted here) are done to ward of evil Prayog done by others.

Chaaturhotra Yagya
A Yagya needs four people to complete it, and when a Yagya is completed by these four people, it is called Chaaturhotra

(1) Hotaa - chants Rig Ved Mantra
(2) Adhwaryu - chants Yajur Ved Mantra
(3) Udgaataa - chants Saam Ved Mantra.
(4) Brahmaa - chants Atharv Ved Mantra

Yagya in Geetaa
In Indian social system, by and large, the word "Yagya" is used for all kinds of auspicious, sacred and spiritual ceremonies. The "Yagya" is also understood as a ritual ie offering oblations to fire God . In this concept Yagya has two parts - Mantra (sound energy) and Havishya (food of gods). The Mantras are phonetic sound symbols for invoking and inviting different Gods and "Havishya" is the food offered for their satisfaction. The essential ingredient of Havishya is "cow's Ghee". The "fire" (Agni ) is said to be the mouth of all gods .Therefore the food is offered to all the gods through Agni's mouth. Thus, all kinds of sacred ceremonies are called Yagya and Yagya is a specific ritual performed to satisfy Divine powers (gods) for healthy, wealthy, happy and peaceful life on earth

The first expression of this Divine word in Geetaa is in its 3rd chapter (Karm Yog – Yog of action) verse 9. Here Shree Krishn has bifurcated Karm (action) in two forms. One is "Yagyaarth" (Yagya + Arth - action performed as Yagya) and another, actions performed otherwise. In Indian spiritual science, 'Karm' is said to be the cause of bondage (to the cycle of birth and death). Arjun was confused and could not understand, how a war could be supportive or instrumental to "Gyaan" (liberation). Shree Krishn told him that any action performed as Yagya liberates the man. Here "Yagya" is the spirit behind actions. Any action performed in Yagya spirit liberates the person and frees from bondage and when performed otherwise binds to the cycle of birth and death.

Four types of Yagya are explained in Geetaa. "Gyaan Yagya" is the highest among all Yagya. Yagya is a spirit, Yagya is a science, Yagya is a ritual, Yagya is a method of self realization and Yagya is a way of living .