A geometry is defined as crossing when at least one vertex of it lies in the interior of the reference geometry (when it's a polygon), or crosses the LineString. An example of a query using this relationship is finding all the bridges crossing the Odra river:

SELECT name,wkb_geometry FROM lines WHERE other_tags LIKE '%bridge%' AND ST_Crosses(wkb_geometry, 
(SELECT ST_Collect(wkb_geometry) FROM lines WHERE waterway='river' AND name='Odra'));