Firstly, you should know that there are NO manufacturing standards for each action and it falls on the maker to class the rod. This makes it very hard to truly put a name on each action, as it does differ greatly across the board
Well in short and simply put, it is the stiffness of the rod or in other words if bent, how quickly it returns to its straight position. Generally speaking a FAST action rod is stiff and will return straight very quickly with most of the curve when bent in the top section of the rod. Whilst generally a SLOW action will straighten out slower and when bent be softer, curving a lot lower down the rod. Actions come in a mix of fast, medium, slow and everything in between, but as mentioned it fall on the maker to decided this.
So what do you need when fishing?
Well starting at Fast action, these are stiffer rods good for game fishing as they don’t bend as much allowing you to get more line back in each lift. Often referred to as having a good back bone for lifting/pulling power. Disadvantages are that some makes are slightly heavier due to a bulkier construction, you can lose some of the feeling through the rod and quite often, they do not cast light weights well but are great with heavier large weights.
A Medium action as such should suit a bit of everything. With a nice bend that aides in casting further whilst still have a bit of back bone for lifting.
A Slow action bends a lot more and can suite a vertical fishing situation like jigging or bottom bouncing.
But AGAIN there is no standard for these actions and no true usage. YOU need to use a few different "actions" as such and find what you like and works for you best.
The Sydney Marine Man