The Torah tells us that every person must have מורא, a sense of reverence and awe, which is to be displayed toward his or her parents (ויקרא י"ט:ג'). The Gemara in Kiddushin (דף ל:), based on this Posuk and a Posuk later in the Torah (דברים ו:י"ג), equates this מורא that one should have towards his parents with the מורא one must have for Hashem Himself. The Rambam (פרק ו' מהל' ממרים הלכה א') labels this Mitzvah to revere one's parents as a very important Mitzvah, comparing it, based on the Gemara, to the Mitzvah of revering Hashem, and documenting (שם הלכה ב') that they are to be compared as well in terms of the severity of the punishment due to one who transgresses these Mitzvos.
The Gemara later in Kiddushin (דף ל"א:) explains exactly what is included in this requirement of מורא, describing that one may not sit or stand in one's parents designated place, nor may one contradict one's parents' views nor even support them (in an inappropriate manner). The Gemara (שם) also indicates that one should not mention his father's (first) name, as that is, apparently, a sign of inappropriately casual behavior and of disrespect; rather, as Rashi (שם בד"ה חכם) explains, one should refer to one's father simply as אבא מורי, my father, my teacher, without saying his name. Although the particular context of the Gemara implies that this is necessary specifically when one is quoting one's father during a public Dvar Torah or Derashah, the Rambam (שם הלכה ג') includes this requirement not to call one's parents by their name (whether they're alive or dead), but rather to refer to one's father as אבא מרי, as part of the definition of מורא under all circumstances. The Shulchan Aruch (יורה דעה סימן ר"מ סעיף ב') rules this way as well; the Pischei Teshuvah (שם ס"ק ב'), however, suggests that if one is specifically asked the name of his parent, he may mention the name, albeit in a respectful fashion.
Rabbi Akiva Eiger, in his commentary to the Shulchan Aruch (שם בד"ה ולא יקראנו), questions this entire ruling that one cannot mention one's parents' names from passages in the Gemara in Berachos (:דף י"ח) and in Shabbos (דף קט"ו.) where certain Amoraim indeed referred to their fathers by their first names. The Aruch HaShulchan (יו"ד שם סעיף ט"ו) cites additional passages which present this question, quoting the Gemara in Gittin (דף י"ד:), in Pesachim (דף קי"ב:), and in Sanhedrin (דף פ.), as well as Pesukim where Yaakov Avinu refers to his father Yitzchak and his grandfather Avraham by name (בראשית מ"ח:ט"ז), and where Shlomo HaMelech refers to his father Dovid by name (מלכים א', פרקים ב, ג, ה, ח ).
The Aruch HaShulchan resolves these difficulties by pointing out that in each of these cases, although the speaker did not call his father אבא מרי, he did use the word אבא (or a form of the word like אבי or אבותי). He then suggests that the aforementioned Gemara in Kiddushin (דף ל"א:) never requires that one refer to his father as אבא מרי except when quoting a Dvar Halacha from him publicly, and that the above rulings of the Rambam and the Shulchan Aruch really mean to forbid only using one's father's name alone, but if one precedes mentioning the father's name by saying אבא, which is a title of respect, it is permissible to mention his name, as all of the above sources imply. He concludes by observing that this must be the rule, because it makes little sense to demand that one refer to one's father as מורי, my teacher, when speaking about general matters where the epithet "teacher" is of no relevance. Only when quoting a Dvar Halacha from one's father must one refer to him as מורי, otherwise אבא is sufficient (along with the name).
It should be noted that the Vilna Gaon rules explicitly (ביאור הגר"א ליו"ד סימן רמ"ב ס"ק ל"ו) that it is permissible to refer to one's father by his first name if he says the name along with the word אבא, which suffices by itself; he supports this ruling by citing some of the same passages in the Gemara cited by the aforementioned Rabbi Akiva Eiger and the Aruch HaShulchan in posing the question. Rav Ovadyah Yosef (שו"ת יביע אומר חלק ב, חיו"ד סימן ט"ו אות ה') adds that it is also permissible for one to say that he is בן, the son of, so-and-so, using his father's name, because saying בן so-and-so is the same as saying אבא so-and-so, and he substantiates this view with several proofs.
The Rambam (הל' ממרים שם הלכה ג') writes that this prohibition to use one's parent's name applies not only when referring to the parent, but even when referring to or addressing someone else who has the same name. In such a situation, one should somehow alter the person's name a little (or use a nickname) so as not to have to mention his parent's name. The Rambam adds, though, that this is required only when the name is very unusual, but if the parent has a very common name, one may use it in referring to or addressing another person as long as the parent is not present. The Kessef Mishneh (שם) comments that he is unsure where the Rambam derived this Halacha from, as it is not mentioned explicitly in the Gemara, and the Tur (יו"ד סימן ר"מ) also objects to this ruling of the Rambam.
Nonetheless, the Shulchan Aruch (שם סעיף ב') does accept this ruling, and the Taz (שם ס"ק ה') explains that when one is in one's father's presence, calling out his name is forbidden because the father will think he's the one being spoken to, and any onlookers may think the same thing. The Shach (שם ס"ק ג'), however, quotes a different ruling from the Derishah: if the father has a common name, one may address someone else with that name even in the father's presence, and if he has an unusual name, one may still use it towards someone else if the father is not present. It is the first view, though, that seems to be more accepted. The Vilna Gaon (שם ס"ק ז) and the Pischei Teshuvah (שם ס"ק ג') cite as the source for this view the comment of Rashi in Gittin (דף ל"ד: בד"ה והלכתא) that the Amora Abayei was really named Nachmoni, but since he was an orphan, he was raised by his uncle Rabbah, whose father was also called Nachmoni, and who thus nicknamed him Abayei, apparently so as not to have to regularly use his father's name (עיין שם בגליון הש"ס לרעק"א).
Rabbeinu Yehudah HaChassid, in his famous Tzavaah (will) wrote (סימן כ"ג) that one should not marry a woman who has the same name as his mother, and that a woman should not marry a man who has the same name as her father, and if such a marriage took place, one of the names should be altered. Rav Ovadyah Yosef (שו"ת יביע אומר שם חאה"ע סימן ז' אות ז') quotes some who say that the reason for this is so that the husband or the wife will not have to constantly use one of their parent's names; he himself believes, though, that this is not the real reason. The Chasam Sofer (שו"ת חתם סופר חלק אה"ע סימן קט"ז) writes, as does the Chochmas Adam (כלל קכ"ג אות י"ג) that Rabbeinu Yehudah HaChassid was concerned there only about three generations having the same name in the marriage, as he writes explicitly in the Sefer Chassidim (סימן תע"ז). The Noda BiYehudah (שו"ת נודע ביהודה מהדורא תניינא חלק אה"ע סימן ע"ט) writes that Rabbeinu Yehudah HaChassid intended this ruling only for his own descendants; Rav Moshe Feinstein (שו"ת אגרות משה אה"ע חלק א' סימן ד') also suggests that a couple may opt to disregard this ruling. Rav Ovadyah Yosef (שם בכל התשובה) quotes numerous other leniencies which allow us to suspend this ruling.