How to add up a column depending on data from another column
The problem is that you are not joining your two tables based on band_id, as a result you are creating a Cartesian product.  When you create this cartesian product, you are combining each row from your placing table to each row in the band table.

For example, if you have data similar to the following:

create table band
(
  band_id number
);

insert into band values (1);
insert into band values (2);
insert into band values (3);
insert into band values (4);
insert into band values (5);

create table placing
(
  band_id number,
  points number
);

insert into placing values (1, 10);
insert into placing values (1, 2);
insert into placing values (2, 1);
insert into placing values (3, 3);
insert into placing values (3, 89);
insert into placing values (4, 45);
insert into placing values (5, 63);


And you query the data without a join on the band_id column:

SELECT
  placing.points,
  band.band_id
FROM placing, band;


You are generating data similar to:

| POINTS | BAND_ID |
|--------|---------|
|     10 |       1 |
|      2 |       1 |
|      1 |       1 |
|      3 |       1 |
|     89 |       1 |
|     45 |       1 |
|     63 |       1 |


See Demo. As you can see band_id = 1 now has every single point value from the placing table even though your really only have points = 10 and points=2.  

In order to get the correct result, you need to JOIN the two tables on the band_id column. Your query will be similar to the following:

SELECT
  SUM(p.points) AS sumpoints,
  b.band_id
FROM placing p
INNER JOIN band b
  on p.band_id = b.band_id
GROUP BY b.band_id
ORDER BY SUM(p.points) DESC;


See SQL Fiddle with Demo. This will give a result of:

| SUMPOINTS | BAND_ID |
|-----------|---------|
|        92 |       3 |
|        63 |       5 |
|        45 |       4 |
|        12 |       1 |
|         1 |       2 |