Responding to 1NT openings can seem complicated at first because of the introduction of Stayman and transfer bids but it’s actually quite simple in essence: 

Balanced hands (without a five card major):

Weak (‘bad’ 8 or fewer points) - Pass 

Invitational (9 or a ‘good’ 8 points) - look for fit with four card major via 2♣ (Stayman) or raise to 2NT without four card major

Game-forcing (10+ points) - look for fit with four card major via 2♣ (Stayman) or raise to 3NT without four card major

Hands with 5+ cards in a major or 6+ cards in a minor suit

Show their suit first via transfer bids, regardless of strength. They will go on to show their strength with their second bid, passing if weak (or correcting 3♣ to 3♦), bidding 2NT if invitational or 3NT if game-forcing. This information is pre-empting the module on responder’s rebids but it’s useful to include here because you can see the sense of the whole system. When responder has shown both their long suit and their strength, the 1NT opener knows whether to play in responder’s long suit or in no-trumps, depending on fit, and whether to play in game or part-score. The partnership gets to play in the best contract and with the strong hand hidden from the opponents

Stayman

Any hand that is invitational or better (so 9+ points), with at least one four card major can find out whether opener also has four cards in that suit by bidding 2♣.  The 1NT opener will bid their four card major if they have one, bid 2♥ (the lower-ranking of two four card suits) if they have both, or 2♦ if they don’t have four cards in either major. When they don’t have either major, now the only place for responder to go is 2NT, which is why you need to have at least an invitational hand to do this.

It’s normally a good idea to find the major suit fit because having four trumps in each hand will usually mean you can also get a ruff in each hand with the ‘spare’ trump when the opponents’ trumps are split 3-2. If you have a 4-3-3-3 shape as responder, though, with the 4 being the major, there is no advantage to having the fourth trump. With three cards in each of the other suits there is no shortage for declarer to ruff in dummy so you may as well play in no-trumps, where you only have to make nine tricks for game, rather than 10 in the major suit contract. So if you have this 4-3-3-3 shape you might just want to raise invitational hands to 2NT or game-forcing hands to 3NT and not bother looking for the major suit fit.