How to delete fields in QGIS?

Recent versions of QGIS support deleting Shapefile fields from the attribute table.

Enable editing and then you can edit the shapefile attribute table and multi-delete the columns you do not need.

In QGIS 3.0 there is also a button to delete attributes in the properties menu. First toggle editing (right click on a layer -> toggle editing) and go to the properties menu (right click -> properties or just double click). Go to the source fields tab, select the attributes you want to delete and click the button on top.

That button is only avaliable for a PostGIS layer. You can delete table columns via the "table manager" plugin.