Count number of occurrences for each unique value

Yes, you can use GROUP BY:

SELECT time,
       activities,
       COUNT(*)
FROM table
GROUP BY time, activities;

select time, coalesce(count(case when activities = 3 then 1 end), 0) as count
from MyTable
group by time

SQL Fiddle Example

Output:

|  TIME | COUNT |
-----------------
| 13:00 |     2 |
| 13:15 |     2 |
| 13:30 |     0 |
| 13:45 |     1 |

If you want to count all the activities in one query, you can do:

select time, 
    coalesce(count(case when activities = 1 then 1 end), 0) as count1,
    coalesce(count(case when activities = 2 then 1 end), 0) as count2,
    coalesce(count(case when activities = 3 then 1 end), 0) as count3,
    coalesce(count(case when activities = 4 then 1 end), 0) as count4,
    coalesce(count(case when activities = 5 then 1 end), 0) as count5
from MyTable
group by time

The advantage of this over grouping by activities, is that it will return a count of 0 even if there are no activites of that type for that time segment.

Of course, this will not return rows for time segments with no activities of any type. If you need that, you'll need to use a left join with table that lists all the possible time segments.

If i am understanding your question, would this work? (you will have to replace with your actual column and table names)

SELECT time_col, COUNT(time_col) As Count
FROM time_table
GROUP BY time_col
WHERE activity_col = 3