Select the minimum value for each row join by another table

Use:

select t2.Name, t2.Dept, min(t1.Record)
from table1 t1
join table2 t2 on t2.ID = t1.ID
group by t2.ID, t2.Name, t2.Dept

or

select t2.Name, t2.Dept, a.record
from table2 t2
join
(
    select t1.ID, min(t1.Record) [record]
    from table1 t1
    group by t1.ID
)a
on a.ID = t2.ID

For filtering add where clause, e.g.:

select t2.Name, t2.Dept, min(t1.Record)
from table1 t1
join table2 t2 on t2.ID = t1.ID
where t1.Date = '11/7/2013'
group by t2.ID, t2.Name, t2.Dept

Please try:

Select
    b.Name,
    b.Dept,
    MIN(Record) Record
from
    Table1 a join Table2 b on a.ID=b.ID
GROUP BY b.Name,
    b.Dept