It is stated Tulasi should not be plucked during Tuesdays, Fridays, Saturdays, Sundays, Dvadasi, and Amavasyai. There is a word called Manvadi Yugadinam, what is its meaning? On which days Tulasi should not be plucked in a year?

It is said that Tulasi must not be plucked on Manvadi, Yugadi, etc. Manvadi and Yugadi are mentioned in the Panchangams. The first day of a Yuga is called Yugadi; as an example the day on which Kali yuga started is Yugadi. Yuga refers to a period; similarly is Manvantaram. The time when a Manvantaram starts is Manvadi; on such days Tulasi must not be plucked. This is not followed strictly; if one is aware of such periods, it is better not to pluck Tulasi on such days.
As mentioned in the query, Tulasi must not be plucked on Tuesdays, Fridays along with days on which Amavasyai, Maasa Pirappu when tharpanam is performed, and dvadasi days.