Student     ApplicationDate
-------     ---------------
Barry       2009-01-01
Barry       2009-01-20
Barry       2009-01-23
Barry       2009-02-01
Barry       2009-02-15
Barry       2009-03-01

Student     Month     Applications
-------     -----     ------------
Barry       1/2009    3
Barry       2/2009    2
Barry       3/2009    1

SELECT
    student,
    DATE_FORMAT(ApplicationDate,'%m/%Y') as Month
    count(id) as Applications
from YourTable
group by ApplicationDate

select
    student,
    year(ApplicationDate),
    month(ApplicationDate),
    count(*) as Applications
from YourTable
group by student, year(ApplicationDate), month(ApplicationDate)

select Student,
       DATE_FORMAT(ApplicationDate,'%m/%Y') as 'Month',
       count(*) as 'Applications'
from tableName
group by Student, month(ApplicationDate), year(AppilcationDate)
order by year(ApplicationDate), month(ApplicationDate), Student

select Student, month(ApplicationDate) ApplicationMonth, count(*)
from table_name
group by Student, ApplicationMonth