I have been given a one year lease on a residential house to sign starting January 21, and ending January 20, 2017, is this correct or should it end on January 21, 2017

Odd that it would state the 20 as an end time. They are usually to the last day of the month with the first month pro rated from the sign time to the end of that month.

But the 20th would be correct. The 21st would be one year and one day as you already will have possession on the 21.

Yes, this is correct for the reasons smoothy noted. Jan 21 to Jan 20 is exactly one year.

As noted that is correct. But I would check the lease and find out when rent is due and what period is covered. For example would the 2 month be due on Feb 21 and the third on Mar 21? If the rent is, instead due on the first of each month, then I would be inclinde to ask the LL to end it on 1/31/17.