Animal sacrifice in Hinduism

The modern practice of Hindu animal sacrifice is mostly associated with Shaktism, and in currents of folk Hinduism strongly rooted in local popular or tribal traditions. Animal sacrifices were part of the ancient Vedic religion in India, and are mentioned in scriptures such as the Yajurveda. Some Puranas forbid animal sacrifice.
Terminology.
A Sanskrit term used for animal sacrifice is "bali", in origin meaning "tribute, offering or oblation" generically ("vegetable oblations [... and] animal oblations,"). 
Bali among other things "refers to the blood of an animal" and is sometimes known as Jhatka Bali among Hindus.
The Kalika Purana distinguishes "bali" (sacrifice), "mahabali" (great sacrifice), for the ritual killing of goats, elephant, respectively, though the reference to humans in Shakti theology is symbolic and done in effigy in modern times. For instance, Sir John Woodroffe published a commentary on the Karpuradistotram, where he writes that the sacrificial animals listed in verse 19 are symbols for the six enemies, with "man" representing pride.
Hindu scriptures.
The Ashvamedha, a ritual in which a horse was allowed to roam freely for a year, then finally sacrificed, is mentioned in the Vedic texts such as the "Yajurveda". In the epic "Mahabharata", Yudhishtra performs the Ashwamedha after winning the Kurukshetra war to become the Chakravartin emperor. The Mahabharata also contains a description of an Ashvamedha performed by the Chedi king Uparichara Vasu, however, no animals were sacrificed in this story. However, sacrifices were common in this ritual. The rulers of the Gupta empire, the Chalukya dynasty, and the Chola dynasty all performed the Ashvamedha.
Agnisomiya was the simplest of all Soma sacrifices in which animal sacrifice played an important part; it required that a goat be sacrificed to Agni and Soma preceding the day of offering of nectar to the gods. In the Savaniya sacrifice, victims were offered throughout the day of offering to Agni. These rituals didn't focus on the killing of the animal but as a symbol to the powers it was sacrificed.
In Bhagavata Purana written in 10th or 11th century, Krishna tells people not to perform animal sacrifices in the "Kali Yuga", the present age. The Brahma Vaivarta Purana describes animal sacrifices as "kali-varjya" or prohibited in the "Kali Yuga". The Adi Purana, Brihan-naradiya Purana and Aditya Purana also forbid animal sacrifice in "Kali Yuga". Some orthodox interpreters of Hindu scriptures, such as Sri Chandrasekharendra Saraswathi, believed that the prohibition in "Kali Yuga" applies only to a few types of animal sacrifices, notably cow and horse sacrifices. Such interpretations justify Vedic animal sacrifice viewing it "as a little hurt caused in the cause of a great ideal" and believing that "the animal sacrificed attains an elevated state".
Animal sacrifice in contemporary Hindu society.
The majority of modern Hindus avoid animal sacrifice, but there are numerous local exceptions. In general, where it is practiced, it will be seen as desired by some deities, but not by others.
Animal sacrifice is practiced in many Shakti temples in the states of Assam, Orissa, Tripura and West Bengal in Eastern India, as well as in the nation of Nepal.The sacrifice involves slaying of goats, chickens, pigeons and male Water buffalos For example, one of the largest animal sacrifice in Nepal occurs over the three-day-long Gadhimai festival. In 2009 it was speculated that more than 250,000 animals were killed while 5 million devotees attended the festival. The Gadhimai festival was banned by the Nepal government in 2015.
In the state of Orissa, every year, animals like goat and fowl are sacrificed before "Kandhen Budhi", the reigning deity of Kantamal in Boudh district, on the occasion of her annual "Yatra"/"Jatra" (festival) held in the month of "Aswina" (September–October). The main attraction of "Kandhen Budhi Yatra" is "Ghusuri Puja". "Ghusuri" means a child pig, which is sacrificed to the goddess every three years. During the Bali Jatra, male goats are offered as a sacrifice to the goddess Samaleswari in her temple in Sambalpur, Orissa. "Bali Jatra" of Sonepur in Orissa, India is also an annual festival celebrated in the month of "Aswina" (September–October) when animal sacrifice is an integral part of the ritual worship of deities namely "Samaleswari", "Sureswari" and "Khambeswari". "Bali" refers to animal sacrifice and hence this annual festival is called "Bali Jatra".
Animal sacrifice is a part of some Durga puja celebrations during the Navratri in the eastern states of India. The goddess is offered sacrificial animal in this ritual in the belief that it stimulates her violent vengeance against the buffalo demon. According to Christopher Fuller, the animal sacrifice practice is rare among Hindus during Navratri, or at other times, outside the Shaktism tradition found in the eastern Indian states of West Bengal, Odisha and Northeastern India, Assam and Tripura. Further, even in these states, the festival season is one where significant animal sacrifices are observed. In some Shakta Hindu communities, the slaying of buffalo demon and victory of Durga is observed with a symbolic sacrifice instead of animal sacrifice.
The Rajput of Rajasthan worship their weapons and horses on Navratri, and formerly offered a sacrifice of a goat to a goddess revered as Kuldevi – a practice that continues in some places. The ritual requires slaying of the animal with a single stroke. In the past this ritual was considered a rite of passage into manhood and readiness as a warrior. The "Kuldevi" among these Rajput communities is a warrior-pativrata guardian goddess, with local legends tracing reverence for her during Rajput–Muslim wars.
The tradition of animal sacrifice is being substituted with vegetarian offerings to the Goddess in temples and households around Banaras in Northern India.
Animal sacrifice is practiced by Shaktism tradition where ritual offering is made to a Devi. In Southern Indian states of Karnataka, Andhra Pradesh, and Tamil Nadu, it is most notably performed in front of local deities or clan deities. In Karnataka, the goddess receiving the sacrifice tends to be Renuka. The animal is either a male buffalo or a goat. 
In some sacred groves of India, particularly in Western Maharashtra, animal sacrifice is practiced to pacify female deities that are supposed to rule the groves. Animal sacrifice is also practiced by some rural communities around Pune to placate deities at temples of Waghjai and Sirkai. In the region around Pune, goats and fowl are sacrificed to the God Vetala. The Kathar or Kutadi community of Maharashtra, while observing the Pachvi ceremony after delivery of a child in the family, offer worship to their family deity, Saptashrungi and also offer a sacrifice of a goat. Following this they hold the naming ceremony of the child on the 12th day.
Ahobilam in Andhra Pradesh, is the centre of worship of Narasimha, the lion-headed avatar of Vishnu, to whom the nine Hindu temples and other shrines are all dedicated. A certain amount of sacrifice of goats and rams is still performed weekly. This is now highly unusual in the worship of Vishnu, suggesting a "transitional state between a wild and unregulated tribal deity and an orthodox form of the god Vishnu".
A popular Hindu ritual form of worship of North Malabar region in the Indian state of Kerala is the blood offering to Theyyam gods. Theyyam deities are propitiated through the cock sacrifice where the religious cockfight is a religious exercise of offering blood to the Theyyam gods.
Animal Sacrifice is practiced by some Hindus on the Indonesian island of Bali. The religious belief of "Tabuh Rah", a form of animal sacrifice of Balinese Hinduism includes a religious cockfight where a rooster is used in religious custom by allowing him to fight against another rooster in a religious and spiritual cockfight, a spiritual appeasement exercise of "Tabuh Rah". The spilling of blood is necessary as purification to appease the evil spirits, and ritual fights follow an ancient and complex ritual as set out in the sacred "lontar" manuscripts.
Method of sacrifice.
Methods for sacrificing range from decapitation, strangulation, to a spike being driven into the heart of the animal.
Jhatka is the prescribed method for Hindu ritual slaughter, however other methods such as strangulation and the use of a wooden spile (sphya) driven into the heart is used. The reason for this is priests see an animal making a noise as a bad omen and the animal making noise indicates that it is suffering. The Jhatka method requires the instant killing of the animal in a single decapitating blow with an axe or sword. Those Hindus who eat meat prescribe meat killed by the Jhatka method.