The Rishonim (see Rashi, Tosfos, Meg 22b) record the minhag of women to refrain from doing work on Rosh Chodesh based on the Midrash that the day was given to them as a holiday in reward for not giving their jewelry for use in the eigel. (Worth thinking about: what does Rosh Chodesh in particular have to do with the eigel?) The Biur Halacha (O.C. 417) writes that the implication of these Rishonim is that the minhag is not simply an optional practice that women can elect to adopt or not adopt, but it is a “minhag tov” that has already been adopted generations ago, and women are obligated to follow the accepted practice.
The achronim discuss whether the minhag applies to both days of a two day Rosh Chodesh or only to the second day. Mishna Berura (417:4) cites the Rokeach who held that the minhag applies to both days, while the Shibolei haLeket and Pri Chadah held it applies only to the second day, but the first day is optional and dependent on minhag hamakom.
Exactly what types of work cannot be done is not clear from the Rishonim - it is hard to imagine that the issur should be stricter than chol hamoed, where many types of work are permitted. Biur Halacha cites R’ Ya’akov Emden’s opinion that only public work is prohibited (i.e. the issur is melacha b’farhesya), and therefore all work done at night is permitted, but the B.H. concludes that he does not know if the minhag as practiced is in accordance with this view. It seems to me that those who prohibit doing laundry with a washing machine (as some practice) do not accept the parameters suggested by RYE, as a private washing machine does not seem to me to be a melacha b'farhesya.