To "sink" means to fall downwards in the water.
What's the difference between "sinking" and "drowning"? First, only people can drown, while both objects and people can "sink". Second, "drowning" means to get caught under the water so that you can't breathe until it becomes dangerous.
When a boat or a ship sinks, it starts to get water inside and gets lower and lower in the water.
The phrase "a sinking ship" is sometimes used as a metaphor to talk about a company, a team, or other organization that is getting worse and worse. The speaker in the example at top said "a boat that's sinking" instead of "a ship that's sinking". One reason for this is that the listener might think that he was speaking about a company or team instead of a real boat if he said "a ship that's sinking".