Long enough for the fielder to demonstrate to the official(s) that he has control of the ball. If he drops it 'during the course of fielding' it's deemed not a catch.
If it is deemed, by the ump, to be dropped in the transfer from glove to throwing motion you won't have held it very long, but it is a judgement call and may go against you. The same for catching a fly ball, its upto the ump. Just don't drop it!_________________________________________________________________________________
A fly ball is a term for when the batter hits the ball and it goes high in the air, and usually makes for an easy out because they are easy for outfielders to catch
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When you are pitching you need to pause for approximately 1 second and it is up to the umpire to call a balk if you don't. When fielding a fly you need to have it in your glove until the umpire sees that you have control. When you transfer the ball from your glove to throw it the umpire will make a decision whether or not you dropped it before the transfer or after. It is a judgement call by the umpire in all cases.