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1) [line 4] EIN NEDARIM U'NEDAVOS KEREIVIN B'YOM TOV
(a) A person may offer a Korban in the Beis ha'Mikdash as a voluntary sacrifice, as it states in Vayikra 1:2. Voluntary Korbanos may be Olos (which are burned entirely on the Mizbe'ach, see Vayikra 1:2-17, 6:1-6), Shelamim (parts of which are eaten, see Vayikra 3:1-17, 7:11-21, 7:28-37) or Menachos (flour offerings, see Vayikra 2:1-13, 6:7-11, 7:9-10).
(b) When a person states, "I pledge an Olah," without singling out a specific animal, his pledge is called a Neder. When he then sets aside an animal with which to fulfill his pledge, and the animal gets lost or dies, he must bring another in its place. If he states, "*This* animal is an Olah," his pledge is called a Nedavah. If the animal gets lost or dies, he has no obligation to bring another in its place.
(c) There is a dispute among the Tana'im as to whether voluntary sacrifices may be offered on Yom Tov, as is recorded in Beitzah 19b. The Tana'im that state that they may be offered argue that: 1. since parts of Shalmei Nedavah are eaten, the Melachos necessary for offering them on Yom Tov are permitted, just as cooking is permitted for the preparation of food. 2. According to some (see RASHI to Beitzah 19a DH Aval Lo Olos), even Olos Nedavah may be offered because the person who offers them benefits from them on Yom Tov by increasing the glorification of HaSh-m. Therefore, slaughtering Olos falls under the category of mi'Toch she'Hutrah l'Tzorech, Hutrah Nami she'Lo l'Tzorech Ochel Nefesh (since they may be slaughtered for a slight Tzorech Hedyot (layman's use), they may be slaughtered even though no preparation of food is involved). (Alternatively, the extra word "la'HaSh-m" (Vayikra 23:41) was written specifically to include offering Nedarim and Nedavos on Yom Tov -- Beitzah 19a, 20b).
(d) The Tana'im that state that they may *not* be offered on Yom Tov rule that even Shalmei Nedavah may not be offered (and certainly not Olos Nedavah) since: 1. they are not considered Tzorech Hedyot at all. Korbanos are brought l'Tzorech Gavo'ah (for supernal use), and the layman only receives his share from on high. Melachos that are performed *solely* l'Tzorech Gavo'ah are not permitted on Yom Tov, since the verse (Shemos 12:16) states "Lachem" (TOSFOS to Shabbos 24b DH Lefi). 2. Alternatively, since the verse (Vayikra ibid.) states "v'Chagosem," only Korbanos such as the Chagigah may be offered. The Chagigah has a set time at which it may be offered. Nedarim and Nedavos, which may be offered on any day, may not be offered on Yom Tov (RASHI to Beitzah 20b DH mid'Oraisa)
(e) All of the above reasons follow the opinion of Beis Hillel. According to Beis Shamai, Nedarim and Nedavos certainly may not be offered on Yom Tov (Mishnah Beitzah 19a)

2) [line 12] TAMID (KORBAN TAMID)
The Korban Tamid, a communal offering (Korban Tzibur) consists of two male sheep that were less than one year old. One sheep was offered in the morning and the second sheep was offered in the afternoon. One tenth of an Eifah (approx. two quarts) of flour was brought with each sheep as a Minchah (meal offering), which was mixed with one quarter of a Hin (approx. one quart) of hand pressed olive oil. One quarter of a Hin of wine for each sheep was brought as a wine libation (Bamidbar 28:5).

3) [line 14] B'MINYANA D'VEI REBBI - among the members of the court of Rebbi (lit. in the count of the study house of Rebbi)

4) [line 15] U'MINAI DIDI HAVU MASCHELEI B'REISHA - and from me they would begin [to hear the arguments in a case] first (see Insights here)

5) [line 21] GEDULAH - greatness, wealth, leadership
6) [line 24] NACH NAFSHEI - he died
7) [line 24] KULHU SHENEI KA'AMRINAN - we are talking about [unequealed Torah scholarship and greatness in leadership in one person, which was unequaled for] all of his years

8) [line 25] IRA HA'YA'IRI - one of the great Torah sages and advisors of King David (TARGUM YONASAN to Shmuel II 20:26). He is Ira ben Ikesh ha'Teko'i (Shmuel II 23:26), who was called ha'Ya'iri because (a) one of his ancestors was named Ya'ir; or (b) Teko'a, his city, was known for its abundant production of olive oil (Menachos 85b). Olive oil at the time was the best source of lamplight, and the word "Ya'ir" means he/it will give light (RADAK to Shmuel II 20:26).

9) [line 27] SHIM'I BEN GERA - The teacher of King Shlomo. He was confined to Yerushalayim (Melachim I 2:36-38) as a result of his rebellion against King David (Shmuel II 16:5-13).

10) [line 28] SHEVNA - a minister of Chizkiyahu ha'Melech who was a great Talmid Chacham. He later turned to evil ways and led a rebellion against the king during the war against Sancheriv. As the Gemara relates (Daf 26b), Sancheriv had him killed. (However, according to Tosfos in Yoma 38b and elsewhere, there were two different people by the name of Shevna.)

11) [line 29] V'HA IKTIL - but he (Shevna) was killed (see previous entry)

12) [line 34] MEIKAF HAVAH KAYIF LEI L'RAV ASHI - he subordinated himself to Rav Ashi

13) [line 36] "LO SA'ANEH AL RIV" - "Do not speak up in a dispute [to pervert justice]" (Shemos 23:2). - Based on the spelling of the word "Riv" ("dispute") in the verse, the Gemara expounds the verse to be saying, "Do not argue with the Rav," i.e. with one who is greater than you. As such, in a case of Dinei Nefashos the younger, less experienced judges are to express their opinions first. (See MAHARSHA and MAHARSHAL, regarding whether this means that the youngest and least experienced judge is to speak first, or whether this means that the judge second to the head judge is to speak first; see also Insights to Sanhedrin 32:2.)

14) [line 39] "VA'YACHGOR GAM DAVID ES CHARBO" - "[David said to his men, 'Gird every man his sword,' and every man girded his sword,] and David also girded his sword" (Shmuel I 25:13)

15) [line 40] SHONEH ADAM L'SALMIDO - a person may teach his student [ways of exonerating a defendant, and ways of finding a defendant guilty, in Dinei Nefashos, and then he may sit with him on the same court and judge a case of Dinei Nefashos] (see RAMBAM, who explains that when a Talmid is a Chacham (wise) and Mevin (understanding), and is lacking only the Kabalah (the knowledge passed down through tradition), his teacher passes the Kabalah to him which he needs for this case and then they may judge the case together)

16) [line 42] HA'TAHAROS VEHA'TEME'OS - cases of determining whether an item is Tahor or Tamei (see Background to Bava Basra 80:13)

17a) [line 2] LI'GEMAREI D'RAV HAVU TZERICHEI - they needed (and were dependent upon) the teachings (information) of Rav
b) [line 2] LI'SEVAREI D'RAV LO HAVU TZERICHEI - they did not need (and were not dependent upon) the logic (way of thinking) of Rav (and therefore they would be able to render their own decision in a case and would not be compelled to follow Rav's decision)

18a) [line 9] HA TISH'AH HAVU - but these are only nine [differences between Dinei Nefashos and Dinei Mamonos mentioned in the Mishnah]
b) [line 10] HA ASARAH KETANI - but the Mishnah mentions ten (the Gemara goes on to explain what the initial question was in assuming that there are only nine differences mentioned)

19a) [line 12] ZAKEN - a very old judge (RAMBAM)
b) [line 12] SARIS - an impotent man
20) [line 13] ACHZARI - a cruel person

21) [line 17] MAMZER - a person born through a certain type of prohibited relationship
See Background to Bava Kama 38:24.

22) [line 20] V'HAVINAN BAH - and we ask about it

23) [line 23] RA'UY LAVO BA'KAHAL - fit to come into the congregation (by marrying a Jewess)

23a) [line 28] MENUKIN B'TZEDEK - (lit. cleansed with justice) totally just
b) [line 28] MENUKIN MI'KOL MUM - clean of any blemish [in their lineage]

24) [line 29] "KULACH YAFAH RA'AYASI, U'MUM EIN BACH." - "You are entirely beautiful, my love, there is no blemish in you." (Shir ha'Shirim 4:7)

25) [line 31] "V'HISYATZVU SHAM IMACH" - "[HaSh-m said to Moshe, 'Gather for Me seventy men from the elders of Yisrael, about whom you know that they are elders of the nation and its officers, and take them to the Ohel Mo'ed,] and have them stand there with you." (Bamidbar 11:16)

26) [line 32] B'DOMIN LACH - with those who are similar to you
27) [line 32] MISHUM SHECHINAH - [perhaps the reason why it was necessary for the judges, in the times of Moshe Rabeinu, to be free of any familial blemish was] because of the Shechinah (i.e. so that the Shechinal would dwell upon them) (RASHI)

28) [line 34] "V'NAS'U ITACH" - "... and they will bear [the burden] with you." (Shemos 18:22)

29) [line 35] KA'CHATZI GOREN AGULAH - like the form of half of a round threshing floor; i.e. a semi-circle (a threshing floor was a circular area surrounded by a low stone wall)

30) [line 36] SOFREI HA'DAYANIN - the scribes of the judges (i.e. of the court)

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