Assuming you actually mean timestamp because there is no datetime in Postgres
Cast the timestamp column to a date, that will remove the time part:
select *
from the_table
where the_timestamp_column::date = date '2015-07-15';
This will return all rows from July, 15th.
Note that the above will not use an index on the_timestamp_column. If performance is critical, you need to either create an index on that expression or use a range condition:
select *
from the_table
where the_timestamp_column >= timestamp '2015-07-15 00:00:00'
and the_timestamp_column < timestamp '2015-07-16 00:00:00';