a symmetry element
If a symmetry operation is the action of turning a tile, the symmetry element is the invisible pin it turns about. It is the geometric thing, a point, a line, or a plane, with respect to which the operation is carried out. You cannot see it in the crystal, but you draw it as a marker: a filled polygon for a rotation axis, a bold line for a mirror.
Each kind of operation has its element. A rotation happens about a rotation axis (a line). A reflection happens in a mirror plane. An inversion happens through a centre of inversion (a single point). A rotoinversion happens about a rotoinversion axis. One element can carry several operations: a 4-fold axis hosts the 90, 180, and 270 degree turns (the 360 degree turn is the identity). So elements and operations are related but not one-to-one.
When you list a crystal's symmetry, say the point-group symbol 4/mmm, every character names an element: a 4-fold axis, a mirror perpendicular to it (the slash-m), and two sets of mirrors. Reading a symbol is reading a list of symmetry elements. The Hermann-Mauguin and Schoenflies notations are just two languages for naming the same elements.
In the symbol 2/m, the '2' is a symmetry element (a 2-fold rotation axis) and the 'm' is another (a mirror plane); the slash says the mirror sits perpendicular to the axis.
Symmetry elements are the axes, planes, and points a symbol lists.
A single element can generate more than one operation (a 6-fold axis gives the 60, 120, 180, 240, 300 degree turns), so 'element' and 'operation' are not the same count.