Attack States
There are various levels of Attacks in this game and they go in increasing priority in this order: Normal Moves, Special Moves, and Hyper Combos. Anytime you perform any of these Attacks, you are in the State of that Attack. So if you perform a Normal Move, you are in a Normal Move State. If you perform a Hyper Combo, you are in a Hyper Combo State.
The reason this is mentioned is because these different Attack States all allow you to cancel into each other in the order of lowest priority to highest priority. For example, at any point that you are in a Normal Move State, during the startup or recovery or a Normal Attack, you can cancel the Normal Attack into a Special Move or a Hyper Combo. At any point during a Special Move, you can cancel your Special Move State into a Hyper Combo. Hyper Combos can also cancel into themselves, but technically that is a Delayed Hyper Combo and those follow their own sets of rules.
It does not matter if the move is connecting on your opponent or whiffing, whether the move has reached its first Active Frame or not or if the move is in recovery: as long as you are considered in that Attack State, you can cancel into a higher priority Attack State.
For example, if you perform a Hard Attack with Morrigan where she does the four hit bicycle kick, you can actually cancel that move at any point, before it hits, during any of the four hits, or even during its recovery into a Special Move or Hyper Combo. This rule applies on hit or whiff. As another example, if you perform Ryu's Shoryuken, as long as that move is going, you can cancel it at any point into a Hyper Combo. This can be on the way up or on the way down and it doesn't matter if the move connected or if it whiffed.
There are quite a few exceptions to the rule, of course. Some moves are just deemed unable to cancel into certain attacks. Wesker's Low Samurai Edge where he shoots at the ground with his gun, for example, cannot be canceled at all into any Special Moves or Hyper Combos. Amaterasu, if she is in Rosary Mode (Whip), her Standing Hard Attack Chain Combo cannot be canceled into Special Moves or Hyper Combos. Also, there are a lot of Normal Moves that just won't let you cancel them early on in their start up. For example, Thor can cancel his Crouching Hard Attack before it hits for extended range on his command grabs. However, you can't cancel it immediately. You can only cancel it after it has traveled a very short distance. There are lots of Normal Moves like this, and, again, you'll have to experiment with your characters to learn which ones have restrictions and which ones don't.
Special Moves are a bit more picky, however, as some moves cannot be canceled the instant they start. For example, Deadpool's Hard Attack Katana-rama can only be canceled into a Hyper Combo after a few frames. It actually isn't able to be canceled the instant you start it. For each of your favorite characters, you will have to learn the specifics of which moves have these exceptions.