This passage is a major source of one of the dominant values in Jewish law and thought - בָֽחַרְתָּ֙ בַּֽחַיִּ֔ים - the primacy of supporting and valuing life.
This is arguably the most important passage from which we can learn about the status of the fetus from a Biblical perspective. The rabbinic Jewish tradition has understood this to mean that if the fighting leads to fetal death, the attacker must pay a monetary amount; but if the pregnant woman herself were to die, then the attacker would receive capital punishment. The distinction between the punishments indicates that the fetus is not seen as a full person [and we are bracketing the "eye for an eye" question of whether the attacker would indeed receive the death penalty in such a case of manslaughter].
There are various rabbinic reflections on the status of the fetus. In this passage from Sanhedrin, Rabbi Yehuda HaNasi describes changing his view through this conversation, and coming to believe that the soul enters the fetus at the moment of conception.
This mishna stands in tension with the previous rabbinic source from Mas. Sanhedrin. Here the fetus is understood to be a part of the mother's body. It is unclear how this text would relate to the notion that the fetus is "ensouled" at conception, as is described in Sanhedrin - would it accept that view, and yet say that still the mother's life is primary and paramount? Or do we see it as a rejection of that view?
Similar to the previous mishna cited, the mishna in Oholot is explicit that the fetus is not a full person prior to birth. Further, it identifies birth as a discrete moment wherein most of the body as emerged from out of the mother.
Responsa Tzitz Eliezer 13:102
It is clear that in Jewish law an Israelite is not liable to capital punishment for feticide.... An Israelite woman was permitted to undergo a therapeutic abortion, even though her life was not at stake.... This permissive ruling applies even when there is no direct threat to the life of the mother, but merely a need to save her from great pain, which falls within the rubric of "great need." Now, is it possible to imagine a case in which there is more need, pain, and distress, than the present one, in which the mother is confronted by the [prospect of a] suffering child whose certain death is only a few years away and nothing can be done to save it?
This is a responsa from an authoritative Israeli rabbi, R Eliezer Waldenberg, from 1975. He was asked about the permissibility of aborting a fetus that would be born with Tay-Sachs disease. His ruling here is important because he extends the permissibility not just in a case where the mother's physical health would be at risk, but the fact that a child born with this devastating disease would cause "pain and distress" - ie emotionally and psychologically - is also a valid reason for abortion.
R Waldenberg, and other modern halachic decisors - write about the topic in other places, and there are certainly those who are more stringent, but it is important to note that positions like this have been seen as mainstream for almost 50 years at this point.