Moysian fits (named for Alphonse "Sonny" Moyse) are simply 4-3 trump fits. These hands can be very tricky to play, but there's no reason to panic. As long as you keep your head, you should be able to handle these fits.
How do we wind up in Moysian fits? Of course, it could be the result of a bidding misunderstanding but I prefer to think it's by design rather than error. Moyse was himself a proponent of opening with 4-card majors in contrast to most modern bidding systems. Be that as it may, he correctly maintained that we are so hung up on looking for 4-4 trump fits that we often miss out on viable contacts with 4-3 fits.
By way of an example, I have chosen a hand that not only illustrates the beauty of a Moysian fit but also exposes the reader to some other popular conventions in the modern game.
My partner, sitting North, opened 1NT with the following semi-balanced hand: ♠ Q J 6, ♥ K 3, ♦ A 6 5, ♣ A J 8 5 2. At this stage, I won't show you my hand and those of the opponents, but rather let you see how the bidding developed from my partner's point of view and his considerations in choosing the final contract.
East overcalled with a Cappelletti conventional bid of 2♣, showing a 6-card suit, not necessarily Clubs. I doubled – a "stolen bid" double indicating I myself would have bid 2♣, Stayman, had East passed. With no 4-card major, partner dutifully bid 2♦. East doubled, indicating that his 6-card suit was, in fact, Diamonds.
I now made another conventional bid, jumping to 3♠. This was the Smolen convention showing 4 Spades and 5 Hearts – had I had 5 Spades and 4 Hearts, I would have bid 3♥ - and 11+ points.
Partner had all he needed to make an informed decision. He correctly assumed that most of my points would be in the majors and reckoned that with such "thin" stoppers in the minors, 3NT was doomed to failure. In contrast, his hand was rich in controls – 2 Aces and a King – which favored a suit contact. Finally having a doubleton Heart meant that if Spades were trumps, the 3 trumps in his hand could be usefully deployed for scoring extra tricks or setting up my Heart suit as needs be. He now made the thoughtful bid of 4♠, which was the final contract.
West led the ♦8 and when partner tabled his hand, my "Thank you Partner" was well meant. Here's the full hand:
3NT, played at most tables, was a non-starter with only 8 tricks available, but 4♠ was a breeze. I won the opening lead with dummy's ♦A and rattled off another 8 tricks in rapid succession: ♥K then 3♥ to the A♥ in my hand; a third round of Hearts, ruffed on the table; Diamond from table ruffed in hand followed by another heart, ruffed in dummy; ♣A, Diamond ruff in hand, 5th heart ruffed in dummy. With 9 tricks already in the bag, I still had the 2 top spades in my hand for a total of 11 tricks.
Thank you partner and Alphonse but it's not always that easy:
Here again, West leads a Diamond which East wins with the ♦A. East continues with the ♦K. If you ruff this, shortening the trumps in your hand will expose you to West's four Spades so you should rather discard a Club which is anyway a loser. Likewise, on the continuation of the ♦Q. There is no point in East's playing a fourth round of Diamonds as you can ruff this in dummy – the short hand. Win the likely Club continuation and cash dummy's 2 high Hearts. Next ♠J to the ♠K in your hand, ruff a small Heart with the ♠A in dummy, setting up your Heart suit, and play the 8♠ to your hand and draw the outstanding trumps. The ♥A and ♥10 give you the last two tricks and the contract.