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1) [line 4] BECHOR ADAM
(a) The Torah requires that every person sanctify the firstborn male of his children, Kosher animals and donkeys, as it is written, "Kadesh Li Kol Bechor, Peter Kol Rechem bi'Vnei Yisrael, ba'Adam uva'Behemah; Li Hu." - "Sanctify to Me every first-born that initiates the womb among the children of Yisrael, among both man and beast; it is Mine." (Shemos 13:2)
(b) HaSh-m wanted to give us merit by having us perform a Mitzvah with the first produce of our efforts so that we should realize that everything is His. A person comes to this understanding when after all his toil he takes these first products, which are as dear to him as the apple of his eye; and gives them to HaSh-m.
(c) Another reason for this Mitzvah is to remember the great miracle that HaSh-m did for us by killing the firstborn of Egypt. (Sefer ha'Chinuch #18)
(d) The first male born to a *mother* must be redeemed by his father as stated in Shemos 13:13, "v'Chol Bechor Adam b'Vanecha Tifdeh." The Bechor must be redeemed when he is one month old by giving five silver Shekalim of Kodesh (each of which weighs 22.8 grams / 0.8 ounces) to a Kohen as stated in Bamidbar (18:16). This applies only if the son was the first issue (i.e. he was not preceded by a Nefel -- stillborn) and was not a Yotzei Dofen (born by Cesarean section). (SEFER HA'CHINUCH #392)

2) [line 4] BECHOR BEHEMAH (BECHOR BEHEMAH TEHORAH)
(a) The Kedushah of Bechor rests on every first-born male of an ox, goat or sheep when it comes out of its mother's womb. Nevertheless, there is a Mitzvah for a person to sanctify it himself (Erchin 29a, based on Devarim 15:19). He must then give it to a Kohen; it may not be redeemed.
(b) If the animal has no Mum (blemish), the Kohen must bring it as a Korban during its first year. After its blood and Emurim (see Background to Yevamos 7:8 and 100:9) are offered on the Mizbe'ach, its meat is eaten in Yerushalayim during the following two days and the intervening night by Kohanim, their wives, children and slaves.
(c) If the animal has a Mum, it must be slaughtered and eaten during its first year. If it developed a Mum after the first year, it must be slaughtered and eaten within thirty days. The Kohen can give it away or sell it, even to a non-Kohen. However, it may not be sold in a meat market or weighed in the usual manner.
(d) Whether or not it has a Mum, it is forbidden to work with a Bechor or to shear it. Any fleece that is removed from a Bechor, even if it came off on its own, is Asur b'Hana'ah. If, upon Shechitah, the animal is found to be a Tereifah, it is Asur b'Hana'ah and must be buried.
(e) Now that there is no Beis ha'Mikdash, a Kohen must care for a Bechor until it develops a Mum. Alternatively, he can sell it, even if it has no Mum, to a non-Kohen, who may eat it after it develops a Mum. It may not be bought, however, in order to be sold for a profit (Sefer ha'Chinuch #393, #445).

3) [line 6] ASURIN B'GIZAH V'AVODAH - see above, entry #2d

4) [line 8] SAFEK MASHKIN
(a) In general, the various levels of Tum'ah are called, respectively: 1. Av ha'Tum'ah, 2. Rishon l'Tum'ah, 3. Sheni and 4. Shelishi. The latter two (Sheni and Shelishi) are called "Velad ha'Tum'ah."
(b) Normally, utensils can only become an Av ha'Tum'ah or a Rishon l'Tum'ah, but they cannot become a Velad ha'Tum'ah. However, the Chachamim decreed that liquids that are touched by a Rishon or Sheni can make utensils become Teme'im (Shabbos 14b, and Tosfos DH Ela b'Mashkin). A utensil that becomes Tamei mid'Rabanan in such a manner is called a Velad ha'Tum'ah.
(c) The reason for the decree of the Chachamim that liquids which acquire Tum'ah from a Rishon and Sheni l'Tum'ah are Metamei utensils, is that there are liquids which are Metamei utensils mid'Oraisa; namely the bodily liquids of a Zav or Zavah. The Zov, semen, saliva and urine of a Zav, and the blood, saliva and urine of a Zavah are all Avos ha'Tum'ah.

5a) [line 13] AYIL KAMTZA - a type of locust
b) [line 14] DACHAN - Tahor, i.e. Kosher

6a) [line 14] MASHKIN BEIS MITBECHAYA - the water that was used and the blood that was found in the slaughtering-area of the Azarah
b) [line 14] DACHAN - Tahor

7) [line 6] NEZIR OLAM
(a) If someone says "I am a Nazir forever" or "I am a Nazir all the days of my life", he becomes a Nezir Olam and is a Nazir till the end of his life. Like a normal Nazir, he is not allowed to 1. cut his hair; 2. become Tamei by touching or being in the same room as a corpse; or 3. consume any products of the grapevine.
(b) A Nezir Olam differs from a normal Nazir in that he may cut off some of his hair at the end of every twelve months when his hair becomes heavy, but he may not shave it off completely. On the day that he cuts his hair, he brings the three Korbanos that a normal Nazir brings at the completion of the Nezirus period, namely a Chatas, an Olah and a Shelamim (see Background to Nedarim 9:5:c). If he becomes Tamei, he brings a Korban Tum'ah, shaves his hair and starts his Nezirus anew (just like any Nazir who becomes Tamei -- see Background to Nedarim 9:5:b). (Nazir 1:2, RAMBAM Nezirus 3:12)

8) [line 7] HICHBID SA'ARO, MEIKEL B'SA'AR - if his hair became heavy (after twelve months), he can cut off some of it (lit. lighten it), with a razor

9) [line 9] NEZIR SHIMSHON
A Nezir Shimshon is a Nazir until the end of his life (see above, entry #7), with the exception that he may come in contact with a corpse (and does not have to bring a Korban for doing so). He is never allowed to cut his hair or drink wine. (Nazir 1:2)

10) [line 14] EIN ECHAD MEHEM NAZIR - [If a person passes by two people and one of the two says, "I shall be a Nazir if that person passing by is a Nazir" while the other one says, "I shall be a Nazir if that person passing by is *not* a Nazir,"] neither one of them is a Nazir

11) [line 14] SHE'LO NITNAH NEZIRUS ELA L'HAFLA'AH - the laws of Nezirus only apply to a distinct utterance (where there is no question or doubt)

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