The definite article is used when specifying a specific instance of a set, where it is expected that both the speaker and the listener know which instance is being referred to, and the instance is not a proper name.
By convention, immediately approaching days of the week are treated, grammatically, as if they were proper names, and thus the definite article is not used.
It will take place on Monday.
This is acceptable, assuming that we are talking about the immediate next Monday that appears on the calendar.
It will take place on Mondays.
This is acceptable, as we are no longer talking about a single Monday, but about all Mondays.
It will take place on Monday after next.
This is not acceptable. We need the definite article, because we are speaking of a specific Monday, that is not a proper name (that is, is not the immediate next one).
It will take place on the Monday after next.
This is now correct. If today is Sunday, March 1st, this refers to Monday, March 9th. If today is Tuesday, March 3rd, it refers to Monday, March 16th.
It will take place on the Monday after next week.
This is also correct, but subtly different in meaning. If today is Sunday, March 1st, this refers to Monday, March 16th. If today is Tuesday, March 3rd, it refers to the same day, Monday, March 16th.
On the week before last...
This is correct, for the same reasons as for days.
On week before last...
This is incorrect, but you might hear it from time to time in spoken English anyway. Spoken English often includes grammatical mistakes. Written English is supposed to be better, but mistakes happen there as well. This particular mistake can be parsed as correct grammar if you treat "week before last" as a noun phrase, but this is definitely non-standard and non-idiomatic.