last and the last

Last week, last month etc is the week or month just before this one.
If I am speaking in July, last month was June.
If I speaking in 1985, last year was 1984.
(Note that prepositions are not used before these time-expressions)

• I had a cold last week. Were you at the meeting last Tuesday?


• We bought this house last year.

The last week, the last month etc is the period of seven days, thirty days etc up to the moment of speaking.
On July 15th, 1985, the last month is the period from June 15th to July 15th.

• I've had a cold for the last week
  (=for the seven days up to today)


• We've lived here for the last year
  (= since twelve months ago)