Convert YYYYMMDD to DATE

The error is happening because you (or whoever designed this table) have a bunch of dates in VARCHAR. Why are you (or whoever designed this table) storing dates as strings? Do you (or whoever designed this table) also store salary and prices and distances as strings?

To find the values that are causing issues (so you (or whoever designed this table) can fix them):

SELECT GRADUATION_DATE FROM mydb
  WHERE ISDATE(GRADUATION_DATE) = 0;

Bet you have at least one row. Fix those values, and then FIX THE TABLE. Or ask whoever designed the table to FIX THE TABLE. Really nicely.

ALTER TABLE mydb ALTER COLUMN GRADUATION_DATE DATE;

Now you don't have to worry about the formatting - you can always format as YYYYMMDD or YYYY-MM-DD on the client, or using CONVERT in SQL. When you have a valid date as a string literal, you can use:

SELECT CONVERT(CHAR(10), CONVERT(datetime, '20120101'), 120);

...but this is better done on the client (if at all).

There's a popular term - garbage in, garbage out. You're never going to be able to convert to a date (never mind convert to a string in a specific format) if your data type choice (or the data type choice of whoever designed the table) inherently allows garbage into your table. Please fix it. Or ask whoever designed the table (again, really nicely) to fix it.

Use SELECT CONVERT(date, '20140327')

In your case,

SELECT [FIRST_NAME],
       [MIDDLE_NAME],
       [LAST_NAME],
       CONVERT(date, [GRADUATION_DATE])     
FROM mydb

In your case it should be:

Select convert(datetime,convert(varchar(10),GRADUATION_DATE,120)) as
'GRADUATION_DATE' from mydb