In order to understand the SUMIF option functionality, Think about the following situation:

We have a table where it shows info of a teacher.
We can watch the fields: Section, Students (number of students) and Status (Approved, Unapproved)
Now, we want to know, how many students approved the course. Previously, we have built a small table containing possible criteria. (see the below image):
We open the corresponding dialog box and fill the data.
Finally we press the Ok button. That is all.

The results are showed next. They are 100 approved students and 100 unsapproved.

Logically, we have used a small table, for this example. but imagine that you must do a similar task, but to a table with 5,000 records. Worrying, don't you?
With this tool, the worrying is over.

In this case we have summed those records classifying in 2 groups "Approved" and "Unapproved", according to the criteria table.
This tool, also, allows you to performance the following opertions: Equal, Greater, Equal or Greater, Less, Equal or Less than, Seaching of text, etc.