Emma's is, as you can probably tell, a chess game - the white queen, representing Emma, has just put the red king, representing Morgan, in check.
In chess, for clarity's sake, a "check" happens when a piece is in a position to capture the opposing king. In this case, the white queen threatens the red king along the left-down diagonal line.
When the king is in put in check, that king's player is compelled by the rules to make a move which will take the king out of check. In this case, the king can move left, right, up... basically anywhere that is not on that particular diagonal.
That's all fairly basic - Emma has put Morgan in check, and now Morgan must respond, but there are plenty of opportunities to do so. Where it gets interesting is when we consider this position not only in terms of the next move, but also how it would develop. Which is to say, it wouldn't.
You see, the white queen cannot put the red king in checkmate and win the game, because if the queen literally corners the red king, cutting off all avenues of escape, it is still legal for the red king to capture the piece that put it in check - but likewise, the red king will never be able to capture the white queen, because that would require the white queen to move next to the red king, which is obviously suicidal.
This position leads to perpetual check, with neither player able to finish the game. Emma cannot checkmate Morgan without getting too close and getting killed - Morgan can't kill Emma because she's smart enough to keep her distance.
However, there is a scenario in which it would be possible for white to checkmate red. It is illegal to move your king into check. Therefore, if another white piece threatened the square where the white queen is standing, the red king cannot capture it, and red would lose.
So it sure is a good thing for Morgan that Emma is working on Heartbreak all by her lonesome. Would be terribly inconvenient if she acquired a partner of some kind.
"Check" in poker means something different than in chess. In chess, allowing for a bit of simplification, it's a threat. In poker, more specifically Texas Hold 'Em as is being played here, to "check" means to pass on betting anything for the round, without folding. If all players check during a round, then the next card will be revealed - in this case, the fourth card, the "turn".
In this game, something extraordinary has happened: the "flop", that is, the three cards revealed after the initial round of betting, when players only know the two cards they themselves have been dealt, contains three of the four queens in the deck.
This means that there are very few hands that are any good - really, it's only the four-of-a-kind with queens that's important. This is because any player can use the cards in the flop (and beyond) to construct their hand, therefore all hands this ante are at least three-of-a-kind. Pairs become full houses, but no flushes, straights, or three- or four-of-a-kind hands that are not queens can be constructed.
So the game is on, and the question is this: who, if anyone, has the queen?
Morgan checks. She does not raise the bet. Does this mean she doesn't have it? Surely if not, she would've folded, right? Or does she just want you to think that? But if she had it, she'd have raised, right? Or is that too obvious?
And there's nothing to prevent her opponent, Emma, from doing the same. A sort of... perpetual checking. (In this case, slightly less perpetual - the game will end after the final card, the river, has been revealed.)
Oh, would you look at that. She did have it. Hope you didn't bet against her.
In this case, the check represents Morgan being put in a dangerous situation, and allowing Emma to gauge how lucky she thinks Morgan is. There's only one way out - for both of them. Whoever has the queen wins - and maybe neither of them do. If Morgan is lucky, she's toying with Emma. If not, she's trying to expose her to the other players by tricking her into betting. Either way, she's given Emma just enough rope to hang herself with. It's a move that cannot be outsmarted, because the intention behind it hinges entirely on something random and unknowable - what card Morgan is holding.
As a final note, we can consider these two checks in concert. Both revolve around a single queen threatening to steal the game, while in actuality provoking repetitive gameplay.
But in chess, the queen belongs to Emma. And the flop doesn't "belong" to Morgan - as I said, it can be used by any player at the table. And we don't see Morgan's two hole cards; it's only when she has a winning hand that we see the queen. It's entirely possible she won't have it. So in chess Emma always "has the queen" and can win, if she knew to play with the flop. In poker, Morgan might have the last of four queens, and wins instead.
Only if Emma learns to play Morgan's game can she win on her own.
That is, unless Morgan was lucky enough to draw the queen. Then, it's a tie... and on to the next game.
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