What is the difference: emergent, submergent floating leaf, and free-floating plants?

Simply, these terms refer to the growth habit of aquatic and wetland plants.  Some plants will have more than one habit.  Updated page on emergent, submergent, aquatic vegetation

Emergents emerge or have a large portion of their shoots, leaves or flowering structures out of the water.  These include the familiar cattails, and also bulrushes, wild rice, sedges, bur-reed and many others.

Emergent Plants: Blue Flag Iris, Tussock Sedge (Carex stricta.), Bluejoint Grass

Submergent plants have most of their structures below water.  Common examples of these would be coontail, milfoils, and many pondweeds.

Submergent Plants: Coontail, Elodea (Canadian Waterweed),
Wild Celery, Water  Strar-grass, Chara

Floating-leaf plants have large floating leaves.  They include the water-lilies, some pondweeds, and American lotus, although the latter often protrude from the water.

Floating-leaf plant - White waterlily (Nymphaea odorata)

Free-floating plants include the duckweeds, common bladderwort and often coontail.  Coontail is sometimes rooted, but it is dislodged easily by wave action and will continue growing in a floating mass, or tangled in with other plants.

Free-floating plant: Least Duckweed (Lemna minor)