OK sorry, that’s the Engineer in me: To an Engineer, or Mechanical Designer, “Normal” means “At right angles to”.
So if you set view to front and make a bone that is vertical, the Y axis of the bone is upright or along the length of the bone from tail to head, following Global Z axis. the bone’s X axis is Global X and the bone’s Z axis is negative Global Y.
Now extrude the top of that bone to the right in the front view. The new bone’s Y axis is still from tail to head and now Global X axis, the new bone’s X axis is now negative Global Y and the new bone’s Z axis is Global Z.
If you normalise those two bones, select both and press CTRL-N > “Local X tangent”, both bones X axes will now align with Global Y, this is useful if you want to rotate the bones about the Global Y axis, you would then set rotations in the X Rotation Transform box for both bones and they will rotate about Global Y.
Confusing? Yes to start with but practice will help you. Just display the armatures axes to see which way round your bones are, in general (I say “in general” before some smarta*** corrects me), The bones Y axis runs from tail to head of the bone and the others obey the Left Hand Rule, where your thumb is Z and points up, your first finger is Y and points away from you and your second finger is X and points across your body from left to right. Just twist your wrist and you will see that all bones obey this rule. Hold your fingers as below for the Left Hand Rule:
Always align your hand so your first finger is pointing from tail to head of the bone and then twist you wrist so your thumb follows bone Z and the rest is easy, until you snap your wrist that is! This rule also applies to Global, Local, View, etc. ad infinitum, the axes in Blender and 3D programmes in general, or at least the ones worth using,
Hope this clarifies things a little.
Cheers. Clock.