Being caught is the most common way of getting out.
A batsman is out caught when a fielder catches the ball directly off the bat, before it has hit the ground.
The ball does not have to come directly off the bat.
It can deflect from the pad on to the bat or from the bat on to the pad and still be caught, so long as the fielder takes the ball on the full.
Also if the ball hits the batter on the hand, below the wrist, then they can still be out caught so long as that hand is in contact with the bat.
But a batsman can not be given out caught off a no-ball.
When a bowler takes a catch off their own bowling, the batsman is out caught and bowled.
And when a wicketkeeper takes the catch the batter is out caught behind.
If the ball gets caught in a batter's pads and a fielder removes it and attempts to claim a catch, the umpire should give the batsman "not out" as the ball is a "dead ball".
If a fielder catches the ball as it crosses the boundary - the batsman is out as long as the fielder does not touch the boundary or the ground beyond it (ie his feet remain inside the boundary, though his hands may be over it).
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