The Mishnah on today’s daf (=page) contrasts between two methods of ritual slaughter of birds – shehitah (ordinary ritual slaughter) and melikah (slaughter performed as part of the Temple service). According to the Mishnah if shehitah was performed by cutting the back of the neck, it is invalid, while if melikah was performed in that manner, it is valid; on the other hand if melikah was performed by cutting the front of the neck, it is invalid, while if shehitah was performed in that manner, it is valid.
Sacrifices brought from fowl are not formally zevahim, a category limited to animals that are slaughtered in an ordinary fashion. Birds are prepared for sacrifice by means of melikah, where the kohen pierces the neck of the bird with his fingernail (see Vayikra 1:14-17 and 5:8-10). The Gemara in Masechet Zevahim (daf 64b) describes melikah as one of the most difficult of the services in the Temple, and explains what was involved.
Rav Zutra bar Tuviah quotes Rav as teaching that the kohen would hold the wings with two fingers and the legs with two fingers stretching out the bird’s neck, and the bird would be killed by means of the kohen‘s thumbnail. According to the baraita, the bird’s body was held in such a way that it was outside the hand of the kohen, and – while holding the wings with two fingers and the legs with two fingers – the kohen would kill the bird with his thumbnail.
According to Rav Ovadiah mi-Bartenura , as well as the Rambam, the kohen would hold the bird in his left hand according to one of the two opinions, and would perform melikah with the thumb of his right hand. This parallels cases of slaughter in the Temple, where both hands are used. The Shittah Mekubetzet quotes Tosafot as suggesting that the entire melikah service was done with the right hand (as depicted in the above illustrations). According to this approach we can easily understand why this service is considered to be the most difficult one, since the bird had to be held and killed with a single hand.