Last week, month, etc (without the) is the week, month, etc just before this one. If I am speaking in July, last month was June; in 2006, last year was 2005.
The last week, month etc (with article) is the period of days up to the moment of speaking. On July 15th, the last month is the period from June 16th to July 15th.
1 – I was ill last week, but I’m ok this week. (NOT
I was ill the last week…)
I’ve had a cold for the last week. I feel terrible.
2 – We bought the house last year.
We’ve lived here for the last year, and we’re very happy with the place.
(Source: Practical English Usage. By Michael Swan. Oxford)