For the past month, we have been reading Sefer Vayikra which describes the sacrifices which are brought on different occasions, first to the Mishkan, the temporary Tabernacle and later to the Beit HaMikdash (Temple) in Jerusalem.

After the Temples were destroyed, we were not permitted to offer sacrifices anywhere else and prayer was instituted in place of the sacrifices.

The Rambam in Moreh Nevuchim 3: 32:6 explains how we have slowly been moving away from sacrifices:

Offering sacrifices, although the sacrifices are offered to the name of God, has not been made obligatory for us to the same extent as it had been before. We were not commanded to sacrifice in every place, and in every time, or to build a temple in every place, or to permit anyone who desires to become a kohen and to sacrifice. On the contrary, all this is prohibited unto us. Only one temple has been appointed, “in the place which the Lord shall choose” (Devarim 12:26); in no other place is it allowed to sacrifice:  “Beware for yourself, lest you bring up your burnt-offerings in every place that you see” (Devarim 12:13); and only the members of a particular family were allowed to officiate as kohanim. All these restrictions served to limit this kind of worship, and keep it within those bounds within which God did not think it necessary to abolish sacrificial service altogether.

Rambam continues: But prayer and supplication can be offered everywhere and by every person. The same is the case with the mitzvot of tẓiẓit, mezuzah and tefillin.

What will happen when the Third Temple is built? Will we go back to bringing sacrifices?

Rabbi J. David Bleich points out in Contemporary Halachic Problems (Volume 1, Part 1):

It is popularly assumed that the synagogue emerged as a communal institution only subsequent to the destruction of the Temple. It is quite true that the synagogue is often referred to as a mikdash me'at, a miniature Temple, but such reference does not connote that the synagogue is merely a replica of, or a replacement for, the Temple which once stood in Jerusalem. Prayer does, indeed, serve as a substitute for the sacrificial order—"Let our lips compensate for bullocks" (Hoshea 14:3)—and the formal order of prayer followed today is patterned after the sacrificial ritual. However, prayer constitutes a mizvah in and of itself, regardless of whether or not sacrifices are concomitantly offered in the Temple.

According to Rabbi Bleich, now that we have organized prayer, we will not have to give it up as it serves a purpose in its own rite.

However, the question still stands- will the sacrifices be back and if so, which kind?

According to Midrash Tanchuma, Buber Emor 19:1: only the Thanksgivings will never cease, meaning the Thanksgiving sacrifice (Korban Todah) and the Thanksgiving prayer which are voluntary will not be cancelled out but there will not be any obligatory sacrifices.

Rav Avraham Yitzchak HaKohen Kook in Olat Re’iya explains what it will be like at the time of Acharit HaYamin, The End of Days:

By the end of time the knowledge of the Lord will extend to the animals, also, as stated by Yishayahu (11:9): "They shall not hurt or destroy in all My holy mountain, for the earth shall be full of the knowledge of the Lord" whereupon this offering, the vegetarian mincha "will be pleasant to the Lord, as in the days of old and as in the former years."

Rav Kook’s view is that eventually we will only bring the Korban Mincha, the vegetarian offering.

We see from here that since we have already moving away from sacrifices, there is a good chance that we will only return to the Korban Todah, a voluntary sacrifice or the Korban Mincha which is vegetarian.