Group By X means put all those with the same value for X in the one group.

Group By X, Y means put all those with the same values for both X and Y in the one group.

To illustrate using an example, let's say we have the following table, to do with who is attending what subject at a university:

Table: Subject_Selection

+---------+----------+----------+
| Subject | Semester | Attendee |
+---------+----------+----------+
| ITB001  |        1 | John     |
| ITB001  |        1 | Bob      |
| ITB001  |        1 | Mickey   |
| ITB001  |        2 | Jenny    |
| ITB001  |        2 | James    |
| MKB114  |        1 | John     |
| MKB114  |        1 | Erica    |
+---------+----------+----------+

When you use a group by on the subject column only; say:

select Subject, Count(*)
from Subject_Selection
group by Subject

You will get something like:

+---------+-------+
| Subject | Count |
+---------+-------+
| ITB001  |     5 |
| MKB114  |     2 |
+---------+-------+

...because there are 5 entries for ITB001, and 2 for MKB114

If we were to group by two columns:

select Subject, Semester, Count(*)
from Subject_Selection
group by Subject, Semester

we would get this:

+---------+----------+-------+
| Subject | Semester | Count |
+---------+----------+-------+
| ITB001  |        1 |     3 |
| ITB001  |        2 |     2 |
| MKB114  |        1 |     2 |
+---------+----------+-------+

This is because, when we group by two columns, it is saying "Group them so that all of those with the same Subject and Semester are in the same group, and then calculate all the aggregate functions (Count, Sum, Average, etc.) for each of those groups". In this example, this is demonstrated by the fact that, when we count them, there are three people doing ITB001 in semester 1, and two doing it in semester 2. Both of the people doing MKB114 are in semester 1, so there is no row for semester 2 (no data fits into the group "MKB114, Semester 2")

Hopefully that makes sense.