surgeon's knot

Properties: A more secure version of the square knot (AKA reef knot, AKA shoelace knot).

Used for: Attach a rope around an object.

figure-eight follow-through

Used for: There are many situations where having a fixed loop on the end of a line is useful.

Properties: It's not difficult to untie, even after taking a hard fall on it. (this is an important rock climbing knot)

Methods for tying: It's useful to know how to tie it two different ways. If you're not tying it around something, it's easier to tie it "on the bight". If you are tying it around something, use the longer method.

Internal notes: Teach this instead of the bowline. They're basically equivalent (students should learn at least one fixed loop; any will do), but IMHO this is easier to teach.

sheet bend

Used for: Attaching the ends of two ropes together.

Properties: The square knot (AKA reef knot, AKA shoelace knot) isn't a suitable knot for attaching the ends of two ropes, this is a better alternative.

round turn and two half-hitches

Used for: Attaching a rope at 90° to a pole / branch / etc.

title

Properties:

Used for:

clove hitch

Internal notes: Don't teach this, except as a basis to teach others. I usually don't find this useful to use by itself.