This is the method of shechitah (ritual slaughtering used today. We are commanded that the method of slaughtering used for animals (for regular eating) must be the same method as used for the altar. This is explained in Talmud Bavli Masechet Chulin 38a. See the Rashi below.
To explain in more detail, the talmud says that we use the method of shechita which is cutting the throat of the animal using a knife to sever more than 50% of both the esophagus and windpipe.
Note that puncturing the animal as with the point of a knife or a spear, invalidates the shechita. Killing or stunning the animal prior to the shechitah invalidates the procedure. The second temple had rings set in the floor to hold the animal in place.
Birds were slaughtered using a different procedure called melikah (using the fingernail to sever the back of the neck). Torah.org cites Vayikra 5:8
8 He shall bring them to the kohen, who shall first offer up that
[bird] which is [designated] for the sin offering. He shall cut its
head [by piercing with his nail] opposite the back of its head, but
shall not separate [it].
but shall not separate [it]: He cuts only one organ [either the esophagus or the trachea]. — [Chul. 21a]
This would not be valid for regular (non-sacrificial) slaughter of a bird for eating.
The details of shechita are explained by the Rambam in The Mishna Torah, Sefer Kedusha, Hilchos Shechita Chapter 1. English
The slaughter which the Torah mentions without elaboration must be
explained so that we know: a) which place in the animal is
[appropriate] for ritual slaughter?, b) what is the measure of the
slaughtering process?, c) with what do we slaughter?, d) when do we
slaughter?, e) in which place [on the animal's neck] do we slaughter?
f) how do we slaughter, g) what factors disqualify the slaughter? h)
who can slaughter?
Ramabam answers these questions in the following halachos.
chabad.org also gives a summary of the method used.
The procedure consists of a rapid and expert transverse incision with
an instrument of surgical sharpness (a chalaf), which severs the major
structures and vessels at the neck. This causes an instant drop in
blood pressure in the brain and immediately results in the
irreversible cessation of consciousness.
כא כִּי יִרְחַק מִמְּךָ הַמָּקוֹם אֲשֶׁר יִבְחַר יְהֹוָה אֱלֹהֶיךָ
לָשׂוּם שְׁמוֹ שָׁם וְזָבַחְתָּ מִבְּקָרְךָ וּמִצֹּאנְךָ אֲשֶׁר נָתַן
יְהֹוָה לְךָ כַּאֲשֶׁר צִוִּיתִךָ וְאָכַלְתָּ בִּשְׁעָרֶיךָ בְּכֹל
21 If the place the Lord, your God, chooses to put His Name there,
will be distant from you, you may slaughter of your cattle and of your
sheep, which the Lord has given you, as I have commanded you, and
you may eat in your cities, according to every desire of your soul.
וזבחת וגו' כאשר צויתך: למדנו שיש צווי בזביחה היאך ישחוט, והן הלכות שחיטה שנאמרו למשה בסיני
you may slaughter… as I have commanded you: We learn [from here] that there is a commandment regarding slaughtering, how one must
slaughter. [Since this commandment is not written in the Torah we
deduce that] these are the laws of ritual slaughtering given orally to
Moses on [Mount] Sinai. — [Sifrei ; Chul. 28a]