How can one tell if one needs to convert in order to be Jewish or if one is already Jewish?
Would one's or one's ancestors' conversion to another religion or their lack of knowledge about Judaism change anything?
According to traditional Judaism, you are Jewish if and only if you yourself have validly converted to Judaism OR your birth mother was Jewish at the time of your birth (Shulchan Arukh Even HaEzer 7:17 and 8:5, Yoreh De'ah 268:6). To determine if your mother was Jewish apply the same rules: either she herself converted to Judaism OR her birth mother was Jewish at the time of her birth. This process recurs at least back to the Revelation at Sinai. NOTE that in order to prove any of these claims you may need to provide more than just hearsay or oral family tradition. If you have a particular concern regarding your own family history, you should speak to your local rabbi for guidance.
As far as conversion to another religion or other lack of Jewish practice, doing so would not take away anyone's status as a Jew. It may create certain obligations related to repentance and losses of certain rights prior to said repentance (see Can a Jew lose their Jewish Status?) but as far as the previous paragraph is concerned, the person is still Jewish.
See this article for more details about the rules and history of Jewish status.
Sorry to spoil the party. DoubleAA's answer talks about "de-jure" Jews, or questionable situations. In practice situation is different.
Whoever is unanimously and publicly accepted as a Jew is a Jew de-facto! As Kiddushin 73a, Rambam Melachim and Shu"A in Even Haezer 2, 5: "משפחה שנטמעה - נטמעה" (Family that assimilated - assimilated.)
I bet, 90% of families (incl myself) can not trace their ascendants back more than 3-4 generations, but in fact it does not matter, as long as those 3 generation openly practiced Judaism we hold their descendants as Kosher Jews de-facto.
As a result, any Reform or other non-Orthodox convert that will see his grandkids observing Torah, will not suffer anyone questioning their Jewishness.