Both terms come from robotics. A rigged character is a chain of bones: upper arm, forearm, hand. Kinematics is how you work out where each part ends up.
Forward kinematics (FK)
You rotate the upper arm, then the forearm, then the hand. Each rotation is relative to its parent, so moving the shoulder moves everything below it. FK is how traditional animators think about arcs: a waving arm or a swinging leg in a walk looks natural because each joint follows its own curve.
Inverse kinematics (IK)
You drag the end of the chain, called the effector or target, and the software rotates the bones to reach it. Put a foot on the floor and it stays there while the hips move. Spine's user guide describes it as setting bone rotations so the tip of a bone touches or points at a target, and its IK constraints work on one or two bones.
Why animators use both
A walk cycle often uses IK for the legs, so feet do not slide, and FK for the arms, so they swing in arcs. A character picking up a cup might start in FK and switch to IK when the hand grabs the cup. Rigs that support IK/FK switching let you do this on one limb.
In 2D tools
Moho has FK and IK with target bones. Spine has IK constraints in its Professional edition, not in Essential. Toon Boom Harmony Premium documents inverse kinematics for cut-out rigs. Rive has an IK constraint. OpenToonz's Skeleton tool has an Inverse Kinematics mode, and Blender's IK constraint works on Grease Pencil armatures.
ExodeUI, which we make, also has bones, weights and an IK constraint with angle and stretch limits, and its bone chains can run in FK or IK mode. It is aimed at characters that live inside app and web UI.