Select info from table where row has max date

SELECT distinct
  group, 
  max_date = MAX(date) OVER (PARTITION BY group), checks
FROM table

Should work.

SELECT group,MAX(date) as max_date
FROM table
WHERE checks>0
GROUP BY group

That works to get the max date..join it back to your data to get the other columns:

Select group,max_date,checks
from table t
inner join 
(SELECT group,MAX(date) as max_date
FROM table
WHERE checks>0
GROUP BY group)a
on a.group = t.group and a.max_date = date

Inner join functions as the filter to get the max record only.

FYI, your column names are horrid, don't use reserved words for columns (group, date, table).

Using an in can have a performance impact. Joining two subqueries will not have the same performance impact and can be accomplished like this:

SELECT *
  FROM (SELECT msisdn
              ,callid
              ,Change_color
              ,play_file_name
              ,date_played
          FROM insert_log
         WHERE play_file_name NOT IN('Prompt1','Conclusion_Prompt_1','silent')
        ORDER BY callid ASC) t1
       JOIN (SELECT MAX(date_played) AS date_played
               FROM insert_log GROUP BY callid) t2
         ON t1.date_played = t2.date_played

You can use a window MAX() like this:

SELECT
  *, 
  max_date = MAX(date) OVER (PARTITION BY group)
FROM table

to get max dates per group alongside other data:

group  date      cash  checks  max_date
-----  --------  ----  ------  --------
1      1/1/2013  0     0       1/3/2013
2      1/1/2013  0     800     1/1/2013
1      1/3/2013  0     700     1/3/2013
3      1/1/2013  0     600     1/5/2013
1      1/2/2013  0     400     1/3/2013
3      1/5/2013  0     200     1/5/2013

Using the above output as a derived table, you can then get only rows where date matches max_date:

SELECT
  group,
  date,
  checks
FROM (
  SELECT
    *, 
    max_date = MAX(date) OVER (PARTITION BY group)
  FROM table
) AS s
WHERE date = max_date
;

to get the desired result.

Basically, this is similar to @Twelfth's suggestion but avoids a join and may thus be more efficient.

You can try the method at SQL Fiddle.