Jewish mother:[1]

According to Jewish law, a person [who did not convert in a Kosher method] is only considered Jewish if his/her mother is Jewish [i.e. direct maternal lineage]. [This means that all his/her direct maternal ancestors are descendants of a woman who was present at the giving of the Torah on Har Sinai, or that someone in her direct maternal lineage converted a Kosher conversion.] One who was born from a Jewish mother is a full-fledged Jew even if one’s father, or someone in the paternal lineage, is/was not Jewish.[2] One whose mother is not Jewish, is not considered Jewish even if their father is Jewish.^[3] [Accordingly, there is no such thing in Halacha as a half Jew or quarter Jew, either someone is Jewish or not. Use of such terms is inaccurate, and does not change the true definition of the person, just as calling an African person a Chinese, does not make him Chinese.] 

[1] Michaber E.H. 4:5 “A Jew who has relations with one of these woman, the offspring follows the mother’s identity.” 4:19 “A gentile who had a child with a Jewess, the child is a Jew”; 7:17; 44:9 [regarding Kiddushin]; Rama O.C. 282:3 [regarding getting Aliyah]; Y.D. 159:3 [regarding Ribis]; E.H. 16:2; 44:9; Shach Y.D. 159:7; Admur 282:8 “A slave who has not been emancipated has the same status as a woman. However, if his mother is Jewish, then he is considered a complete Jew for all matters.”; Admur Hilchos Ribis 79; Rambam Hilchos Issurei Biyah 12:7; 15:3-4, 6; Rebbe Yochanon in Yevamos 17a and 23a and 45b based on Devarim 7:3-4; Mishnah Kiddushin 66b, 68b; Ramban on Vayikra 24:10

Did we ever follow paternal lineage and not maternal lineage? According to some sources, prior to the giving of the Torah, paternal lineage was followed to determine the identity of a Jew, and not maternal lineage. [See Sifra Toras Kohanim Emor 14 and Chochmei Tzarfat brought in Ramban on Vayikra 24:10] The Ramban ibid negates this suggestion, and claims we always followed maternal lineage, even prior to Matan Torah.

[2] Michaber and Rama ibid; Rambam ibid; Gemara ibid

Conversion: Some Poskim rule that the daughter of a gentile father and Jewess requires conversion. [Yearos Dvash Derush 17, brought in Pischeiy Teshuvah E.H. 4/1; Avnei Nezer 16 on Tosafos Yevamos 16b] Practically we do not rule this way. [Avnei Nezer 16; Yeshuos Yaakov 4/8; Beis Yitzchak 26; Achiezer 21; Chazon Ish 6/7]

[3] Michaber and Rama ibid; Rambam ibid; Gemara ibid

Other opinions: The Midrash records the opinion of Rebbe Yaakov Kfar Nevorai who held we follow paternal lineage, and not maternal lineage. His teacher, Rebbe Chagaiy negated this opinion as inaccurate and told his student, Rebbe Yaakov, that he deserves lashes for making such a statement. [Bereishis Raba 7b] This Rebbe Yaakov Kfar Nevorai is considered a controversial figure in the Talmud, and some considered him a sinner and heretic.