In the book of Ruth it never openly states that she converts and in fact she is referred to as Ruth the Moabite throughout the text. At what point, chronologically, did she in fact convert?
With regard to when she converted, both Ibn Ezra and Ralbag seem to believe that she converted before she married Machlon, being that we don't find any mention of her conversion afterwards and Boaz certainly would not have married her otherwise.
Akeidas Yitzchak condemns this approach, though, since Naami explicitly tells her to return to her people, and why would Naami try to convince her to retract her conversion? (Although Ralbag does mention that this was because Naami wanted to see if her conversion was legitimately sincere.) Akeidas Yitzchak therefore believes that Ruth converted on the way with Naami.
As for why the text continues to refer to her as "המואביה", first of all, as @Alex has noted in the comments to the other answers, it is not uncommon to find someone referred to by the country of origin even if they are no longer associated with it. Second, it is important to keep in mind the purpose of Megillas Ruth in the first place. Abarbanel writes that Shmuel Hanavi wrote Ruth after he anointed David to highlight David's ancestry and to show that he came from noble stock. The Chida writes that Shmuel wrote Megillas Ruth to counter the claims that arose later against David's kingship saying that he was unfit being that he descended from a Moabite which the Torah prohibits entering our nation. The famous response to this was "מואבי ולא מואביה" (only male Moabites are prohibited to marry). Shmuel wished to show how even though Ruth was a Moabite, she was a righteous woman and earned the respect and marriage of the great shofet Boaz. (Chida refers to this as Shmuel's "p'sak" that David was legitimate.)
With that in mind, it is quite possible to say that the reason the text continues to stress her Moabite origins is precisely to counter that claim that was common at the time of its authorship. In other words: even though she was a Moabite, look how great she was and how Boaz accepted her etc. However, Alshich writes the opposite. When he sees the text refer to Ruth as "מואביה", he explains that even though she had already converted, she still retained some of the nature of her original upbringing in the way she conducted herself. He explains at length what it was that she did that was "Moabite in nature".
There are indeed a lot of opinions on when Ruth underwent conversion. Rashi (to Ruth 1:12) states that she (and Orpah) were still gentiles when they were on the road back to Eretz Yisrael (indeed, we derive from their conversation the halachah that we are to attempt to dissuade a prospective convert (Yevamos 47b)). On the other hand, Zohar Chadash strongly takes issue with the idea that Machlon could have married a non-Jew; it explains that basically he coerced her into converting (and then, only after her husband's death, did she accept Judaism voluntarily).
I once suggested (Kovetz Haoros Ubeurim Oholei Torah, no. 982, page 76), based on some other Midrashic statements, that at the time of her marriage Ruth was halachically not yet an adult, because she was an ailonis (a woman who never undergoes the normal bodily changes of puberty) but didn't exhibit the symptoms thereof (listed in Rambam, Hil. Ishus 2:6); in that case she reaches halachic maturity only at age 35 (ibid. 2:4). So Machlon had her converted as a child (as per the halachah - Shulchan Aruch, Yoreh De'ah 268:7), but once she reached halachic adulthood she had the opportunity to renounce her conversion (ibid.) - and that's why Naomi had to point out to her the full ramifications of her decision, so that her acceptance of the mitzvos would be a conscious one.
(And once this was accomplished, then it retroactively validated her conversion as a child - just as is true today when a child is converted and then continues to keep mitzvos after bar/bas mitzvah; thus, no further conversion ceremony was needed when they arrived in Eretz Yisrael, and indeed none is recorded.)
In this shiur Rabbi Daniel Glatstein makes the point, based on the Tzafnas Paneach, that it is possible that Rus and Orpah converted before they married but that their conversion was in doubt. He points to Rambam issurei biah 13:14-15 which states that if there is reason to suspect that the conversion was for reasons beside belief (money, power, love etc.) then the conversion has to be investigated. He goes on to say that Naomi probed the motives behind the conversion of her daughter in laws during her trek home as did Boaz during his initial meeting with Rus. Once Rus came to Boaz at night this clinched his belief that her conversion was for the right reasons and he immediately proceeds to "redeem" her.
I believe that it is the Malbim who says that when Ruth Says Ameikh Ami Elokayikh Elokai (your nation is my nation your God is my God) it means that Ruth has taken on keeping the mitzvoth and at this point begins living as a Jew.
It is clear (at least from the p'shat of the text)that she didn't go through any formal conversion process that we would recognize today. This book is often used by people who want to undermine modern halakhic standards of giyur.
While many believe Ruth converted when she said in 1:16, that “your G-d is my G-d,” Gersonides, also known as Ralbag interpreted chapter 1 to be saying that Orpah and Ruth converted to Judaism prior to marrying Mahlon and Chilion, and the proof is that her conversion is not mentioned for Boaz. Naomi even later tested them to see if they have converted. Thus, the rabbi disagreed as to when Ruth converted.