Summarize grouping by year and month
This is what pivot tables are for.
I have created an example using your data.
First, add a column to format the date as your monthly string, eg "2009-01" with the formula:
TEXT(A2, "YYYY-MM")
Then highlight the data, and choose "data" > "pivot table report..."
For Rows, select "month"
For Values, select "amount"
TADA! That's it!
For a quick overview of pivot tables, see this Google blog post.
You can do it with an array formula, i put the formula in A2 and in A1, i put the month number:
=ARRAYFORMULA(SUMPRODUCT(1*(MONTH(Sheet1!$A$2:$A$6)=A1)*Sheet1!$B$2:$B$6))
See the doc in action in Google Spreadsheet
[EDIT] If you want both year and month:
=ARRAYFORMULA(SUMPRODUCT(1*(YEAR(Sheet1!$A$2:$A$8)=A2)*(MONTH(Sheet1!$A$2:$A$8)=B2)*Sheet1!$B$2:$B$8))
The answer of matt burns and Cœur is pretty accurate and I would like to add a comment (I did not add it on the comment section as it is quite long)
You may want to use this formula instead:
DATE(YEAR(A2);MONTH(A2);1) to group by year & month.
This way you will keep values as dates and therefore be able to apply any kind of date formatting or sorting in your pivot table.
Explanation you can skip:
Instead of TEXT(A2, "YYYY-MM") converts value in text and therefore remove freedom about further date manipulation.
DATE(YEAR(A2);MONTH(A2);1) remove the day and time from the date so for example 2018/05/03 and 2018/05/06 both become 2018/05/01 which is still a date which can then be grouped together.
By applying a format "YYYY-MM" directly on the pivot table you reach the same result.
in addition to contribution from JMax, let me present the following formula using the QUERY function ...
=query(index('Sheet3'!A:B); "select year(Col1)+(month(Col1)+1)/100,sum(Col2) where Col1 is not null group by year(Col1)+(month(Col1)+1)/100 label year(Col1)+(month(Col1)+1)/100 'mydate',sum(Col2) 'amount' ")
Notes: I have assumed the original data is in Sheet3 and my result presents the year and month combinationas yyyy.mm instead of yyyy-mm I have created an illustration:
