There are a few different ways to move your content between boards...
Drag content into a board
Move content to a higher board
You can drag content out of your current location and into a higher board using the breadcrumb navigation in the top left of the screen.
Drag your content
Hold it over the top breadcrumb navigation for 1 second. This will open the relevant board
Drop your content into the destination board
Move content between boards quickly
Drag and hold any content over a board's tile to open it, so you can place your content anywhere.
Cut and paste
1. Right-click the item you want to move and select "Copy" from the menu
2. Right-click again in any board and select "Paste" from the menu
Why can't I prompt the right-click menu?
Is the card you're trying to right-click on currently in edit mode? You can tell it's in edit mode by the black border, and the appearance of a blinking cursor inside, like this...
If this is the case, you can click off of the card to de-select it, and then hold down the Command key (MacOS) or Control key (Windows) while right-clicking.