Reading or having reads is the act of predicting your opponent's next attack and perform a specific action that will counter it. e.g. if you read that your opponent will do a backdash to escape your tick throw you can use another long range attack right after your jab that is specially designed to catch backdashes.
Reads are opposed to reactions because the latter require to actually see the visual cue of the opponent's action before pressing any button. That doesn't mean however that reactions are devoid of any kind of anticipation or expectation: many (if not most) great "reactions" of pro players are actually the result of the player actively waiting for a specific action done by the opponent to trigger the counter-attack. They might also use techniques like buffering to make the reaction easier.
As opposed to regular reads, hard reads imply to do a usually risky action way before the opponent even started to attack, often in a neutral situation or without any immediate conditioning. Hard reads are usually required for characters such as grapplers that have slow normals or can be heavily punished on whiff. Example: Rose doing a Soul Throw on the opponent's wake-up to catch a neutral jump can be considered as a hard read because the special move only catches airborne opponents and is heavily punishable on whiff.
The ability of consistently performing hard reads and being in the opponent's head is called yomi in Japanese.