Point vs direction

A point has w = 1, so a translation moves it. A direction has w = 0, so the same translation leaves it in place.

Read the product from right to left. Rotation touches the arrow first. Translation sits in the last column. The matrix multiplies that column by w.

Translate X 1.15 Translate Y 0.45 Rotate Z 28°

w = 1
w = 0

The rightmost column is the translation. A point adds that column. A direction multiplies it by zero. Rotation still turns both arrows. The dark dot is the origin. The origin does not move.

This is the same 4×4 matrix for both arrows. Read each row from left to right.
—	—	—	—
—	—	—	—
—	—	—	—
—	—	—	—

Where this is used

A surface normal is a direction. It tells a light which way a face points. Moving the object does not change that direction. Lighting in games and films depends on this.
In a physics simulation, a position is a point and a velocity is a direction. A shift of the world moves the point. It does not add to the velocity.
On a map, a location is a point and a compass heading is a direction.
Graphics hardware stores a point and a direction in the same kind of vector. The extra number, w, marks which one it is.

Common mistakes

A surface normal or a velocity is a direction. Translating it as if it were a point adds a shift that should not be there.
Dropping w treats a point and a direction as the same kind of vector.
Rotation still turns both arrows. A turn does not treat the point and the direction differently.
The rightmost column is the translation. A point adds that column. A direction multiplies it by zero.

Order of transforms · Next: shear