Firstly, well done for realising that you need to use the correct column type for dates. It avoids formatting problems and takes up less space.
As you already have your table you have a few options when it comes to converting that column.
1. You could drop the table entirely, recreate it and re-populate it
2. You could add an extra column of the correct type then use the technique from solution 1 to populate the new column. E.g.
UPDATE tablename SET newColumn = CONVERT(date, oldColumn,103)
(You might want to delete the old column afterwards). You can do this with ALTER TABLE or via Management Studio
3. You could create a new table from the contents of the old one e.g.
SELECT column1, column2, column3, CONVERT(date, oldColumn,103), .. etc
INTO NewTable FROM OldTable
(NB you must list the other columns) and then drop the old table
If it was me I would probably go with option 1 unless you really have lots of data you can't easily reproduce
[EDIT - added ,103 as the format to use on CONVERT]