Try bending a sheet of paper - it's easy. Roll the paper into a tube and try bending it again - it's a lot harder. How easy it is to bend a sheet of paper or an arrow shaft depends on its stiffness. The accepted (Easton derived) standard for the stiffness of an arrow shaft is its static spine. The shaft is supported at two points a specified distance apart and a specified weight hung at the mid point. The amount the mid point of the shaft drops from the horizontal determines the shaft spine. The lower the stiffness of the shaft the more it sags and the larger the measured deflection. Given the support spacing and the weight hung the static spine depends on the elasticity of the shaft material's) and the materials' geometries. In the case of multi-layer arrow shafts (carbon/aluminum) the stiffness also depends on the bonding between the different layers. The geometrical factor is the inside diameter and thickness of each material layer. You can find in any engineering handbook a 'beam' formula which in theory would allow you to calculate the static spine for an arrow shaft from the material and geometrical properties. With arrows having a non uniform cross section (barreled shape) you can still have a measured static spine, though in this case you have to define where the arrow supports are with respect to the varying shaft geometry. The spine of an arrow shaft (excluding external factors like temperature) never changes unless the arrow material properties change (e.g. aluminum arrows stiffen over time with use as the crystalline structure alters) or the shaft construction changes (cracks, debonding).

When an arrow is being shot then the term dynamic spine is often used. Spine in this context has nothing to do with static spine i.e. stiffness. What is being talked about is how much the arrow bends. How much the arrow bends depends on many factors (shaft stiffness and length; pile, fletching and nock weights, string force and bracing height etc. etc. etc.). So if say you see the expression "increasing pile weight reduces spine" what is meant is that increasing the pile weight will result in the arrow  bending more (the actual 'spine' of the arrow shaft of course remains exactly the same). The terms 'Weak' (bends more) and 'Stiff' (bends less) are often used as an alternative to dynamic spine. So the expression 'Adding fletching increases arrow stiffness' has nothing to do with arrow stiffness, it means that adding fletching will reduce the amount the arrow bends when shooting it. Confusing in nit! Unlike the static spine case there is no simple equation to describe the bending of an arrow while being shot. One of the assumptions in deriving the 'beam' equation mentioned above is that there are no compressive or tensile (stretch) loads on the arrow. When an arrow is being shot you have the string force acting up the arrow's backside creating a compressive load in the shaft so the simple beam formula goes out the window. There are a number of number crunching approaches to modeling this sort of situation. The usual approach is to break the arrow shaft down into lots of small lengths (finite elements) and work out what happens with each small section in relation to the sections either side and build up a composite picture from the bits. Finite element analysis as it is termed is as much an art form as a science.
 

The third usage of spine you occasionally come across is meaning 'direction of flight' of the arrow. Suppose with a set of  arrows you shoot a particular group on the target. If you replace the arrow set with one where the arrows bend more when being shot than the arrow group will shift to the right (RH archer). Sometimes you come across an expression like 'So and so makes the arrow weaker/decreases spine' when what is meant that so and so tends to make the arrows fly to the right.