Page and column breaks are used to force the report engine to make a jump to the next page or column. A column break in a single column report has the same effect as a page break.
In the Design tab, they are represented as a small line. If you try to resize them, the size is reset to the default, this because they are used just to set a particular vertical position in the page (or better, in the band) at which Jaspersoft Studio forces a page or column break.
The type of break can be changed in the Properties view.