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I have some typical gryphons in my world. Bird head and wings, cat bottom half. Four legs and two wings.
How crucial is a bird tail for flight? Could my gryphons be good flyers with just having a cat tail instead, or are the two bird wings good enough?
No magic should be involved.
If your world is more or less like ours, and there is no magic, then your typical gryphons will never be able to fly. By typical I'm assuming you mean the size of big cats. There's a reason why large birds don't fly, or can only fly a bit by launching from high perches, or fly very clumsily, or rely mainly on soaring on thermal currents. It has to do with the strength of muscles v. the weight of those muscles and the bones that support them. Even forgetting those inconvenient facts, I foresee stability problems, as a typical cat-like animal would have a relatively big and heavy head on one end and bulky, heavy hindquarters on the other, which would make it very difficult for it not to lose balance if there's only one pair of wings in the midsection.
I would advise that you gave the physical realism of the setting a big handwave and just go with the typical gryphons as you imagined them, unless you want them to do something specific with their bird tails. Gryphons belong in the fantasy genre anyway, so their presence should signal the reader that they shouldn't expect scientific accuracy.
Firstly, flight does not actually require a tail.
Many of the pterosaurs lacked tails, or only had very rudimentary ones.
Further, some arboreal animals use their tails as in-flight rudders and counter-balancers. For one famous example, consider squirrels.
So you could use the feline tail on your gryphon to act either as a counter-balance (shifting weight) or aerodynamically (perhaps it is flattened). And there is no reason why your gryphon could not use both methods.
Tails are one solution to the stability problem; other solutions exist in nature. Bats have more mobility in their wings due to the wings being stretched between their "fingers" instead of attached to their arms, giving them their characteristic erratic flight pattern which is actually more agile than birds. Insects often have an extra pair of wings, or in the case of flies, specialized stabilizing organs where their back wings used to be. Bees beat their wings in a figure-eight pattern that effectively lets them "flap" twice for every wingbeat, once in the front and once in the back, letting them fly effectively and even hover despite their small wingspan-to-body-size ratio.
Gryphons are typically depicted with bird wings, so my guess is that they wouldn't be quite as agile in the air as birds without a feathered tail, but on the other hand they probably don't need to be. My guess is that a realistic gryphon would stalk its prey like any large cat, then pounce and use its wings to gain extra speed and mobility during said pounce. A large (lion-sized) gryphon wouldn't be able to achieve true flight anyway due to the square-cube law.
A small gryphon, the size of a housecat, might be able to fly. If you give it bat-like wings instead of bird-like ones it may be more agile in the air and won't need a tail for steering.
Also worth noting that the cat's tail is already pretty good at mid-air stabilizing; that's what it's for. It uses its long length to counterbalance the cat's motion in the air instead of aerodynamic properties, but it could be used to steer during a jump.
Taking from the fantasy world, Pegasus doesn't have a bird tail. Taking from reality, bats don't have bird tails and fly superbly. A whole lot of flying insects also do not have stabilizing tails.
Most flying vertebrates had ancestors with a long tail. Early pterosaurs (like Rhamphorhynchus), early birds and early bats all started out with a long bony tail. So the 'proto-flyer' in each group could have had something just like your cat's tail.
Their descendants then stiffened the tail. Then their descendants shortened the tail.
The reason for the shortening of the tail in all these groups is that they are sacrificing stability in the air for mobility. Long tails enable very stable flight. Short flexible tails let you be manoeuvrable. Thus any creature with a lifestyle which means it needs to chase prey in the air, dodge predators in the air, zigzag around tree trunks when flying in a forest, or such things will end up with a short tail.
Also, you can use the short tail as an aerofoil to generate lift. I suspect (don't know for sure) that is more tricky with a long tail, where the drag it generates may outweigh the lift.
