user13744: The rule says:
Delivering a double-check is worth +5, and triple-checks are worth +15
Is a double check when I am 

checking two players at one move
checking one player with a normal double check
checking a player that is already under check

Counting that there is a "triple check" either the rule one or three should be true because you can not give a triple check to one player by only your move. However, I am not sure if there are points for the normal double-check.

user14471: Double Check in 4 man chess basically means  checking  two kings in one move.
Same goes for triple check.
PS. It's not worth the points so don't go for it unless you win because of it.

user33649: Double Check
As @A.Shetye has pointed out, Double check means checking two kings at once in your single move.
More discussion on the topic is mentioned in this chess.com post - Double and triple checks 4pc
Double Checkmate
Well, in fact, in 4-player chess, you can even "Double Checkmate" your opponents. Have a look at a recent game I've played. I've checkmated both the kings at once xD

Here's the game link - https://www.chess.com/variants/4-player-chess/game/31704333/154/4

13744: <p>The rule says:
<strong>Delivering a double-check is worth +5, and triple-checks are worth +15</strong></p>

<p>Is a double check when I am </p>

<ul>
<li>checking two players at one move</li>
<li>checking one player with a normal double check</li>
<li>checking a player that is already under check</li>
</ul>

<p>Counting that there is a "triple check" either the rule one or three should be true because you can not give a triple check to one player by only your move. However, I am not sure if there are points for the normal double-check.</p>
 14471: <p>Double Check in 4 man chess basically means  checking  two kings in one move.
Same goes for triple check.</p>

<p>PS. It's not worth the points so don't go for it unless you win because of it.</p>
 33649: <h2>Double Check</h2>
<p>As @A.Shetye has pointed out, Double check means checking two kings at once in your single move.</p>
<p>More discussion on the topic is mentioned in this chess.com post - <a href="https://www.chess.com/forum/view/chess960-chess-variants/weird-double-and-triple-checks-in-4-player-check" rel="nofollow noreferrer">Double and triple checks 4pc</a></p>
<h2>Double Checkmate</h2>
<p>Well, in fact, in 4-player chess, you can even &quot;Double Checkmate&quot; your opponents. Have a look at a recent game I've played. I've checkmated both the kings at once xD</p>
<p><a href="https://i.stack.imgur.com/d5YWP.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/d5YWP.jpg" alt="Double checkmating - 4 Player Chess" /></a></p>
<p>Here's the game link - <a href="https://www.chess.com/variants/4-player-chess/game/31704333/154/4" rel="nofollow noreferrer">https://www.chess.com/variants/4-player-chess/game/31704333/154/4</a></p>
