1) Haircuts and shaving [any hair on the body] is prohibited [for men] during Sefira. (Shulchan Aruch Siman 493:2)
If one works at a job where not shaving will cause him a monetary loss, it is permitted. However, if one is simply embarrassed to go unshaven, and is worried that his co-workers will poke fun at him, but he will not incur a monetary loss, the prohibition stands. (Psak of Rav Moshe Feinstein Zatzal in Igros Moshe Orach Chaim Vol. 4 Siman 102 and Choshen Mishpat Vol. 1 Siman 93. See She'Arim Metzuyanim B'Halacha on the Kitzur Shulchan Aruch Siman 122:3 for a lengthy discussion with other sources about this)
In a year when Rosh Chodesh Iyar falls out on Shabbos (such as this year) it is permissible to shave on Friday (tomorrow), L'Kovod Shabbos and L'Kovod Rosh Chodesh, no matter which half of Sefira you observe. (Mishna Berura 493:5)
2) Many Poskim are lenient in regard to women, and maintain that women may indeed shave any of their hair during Sefira.
However, the prevalent custom is for women to be stringent and not cut the hair [on their head] during Sefira unless doing so in preparation for immersion in a Mikvah or in cases when the hair is too long and thus not able to be covered properly according to Halacha.(See Shu"t L'Horos Nosson Vol. 2 Siman 32. This is also the opinion of Rav Shlomo Zalmen Auerbach Zatzal and yblc"t Rav Shmuel Wosner Shlita)
The above Halacha refers to hair of the head only. However, women may shave their legs, pluck their eyebrows, and have electrolysis done L'Chatchilah during Sefira. (See Hilchos Bein HaMetzorim from Rav Shimon Eider Zatzal page 4 in the name of Rav Moshe Feinstein Zatzal and Sefer Bein Pesach L'Shavuos page 241 where he quotes Rav Shlomo Zalmen Auerbach Zatzal regarding the eyebrows and electrolysis that they aren't considered "shaving" )
The prevalent minhag is to not give haircuts to children under Bar/Bas Mitzvah during Sefira, unless absolutely necessary.
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