Plants are placed into two simple kingdom subsections: nonvascular and vascular plants. The big distinction between the two groups is that vascular plants have vascular_tissues: tissues that conduct water and dissolved food in the plant body. The vascular tissues include xylem, which moves water and minerals upward through the plant and phloem, which moves dissolved food downward in the plant.
Nonvascular plants lack these tubes, and therefore must be small and thin so that each cell can acquire its needed materials via passive_transport from the plant's surface, or from other cells within the plant that are located near the surface.
Regardless of whether a plant is vascular or nonvascular, it has structures that (1) anchor it into the soil, (2) provide physical support for its photosynthetic structures, and (3) specialize in performing photosynthesis.
The difference is that in nonvascular plants these structures are called root-like, stem-like, and leaf-like because they lack vascular tissues. True roots, stems, and leaves are those body structures that contain vascular tissues.
Equisetum (the horse_tail) grows two types of shoots from a single rhizome (a rhizome is an underground horizontal stem). The first shoot to appear is the reproductive shoot, which produces a single terminal_strobilus. The second shoot (the sterile_shoot) appears later, and is responsible for photosynthesis. Photograph 50A shows a comparison of the two types of shoots. Your slide (50C) shows a mature male gametophyte with antheridia and rhizoids (to act as anchors).