Learn English – a fork’s single point called

I've heard the term cubicles for these compartments.

This is called packaging tape, or packing tape. The verb used would usually be simply to tape. If the speaker wanted to be more specific, he could say "I taped the box shut with packaging tape".

"Scotch tape" is generally used in the United States for a much lighter variety of tape that is mostly suitable only for taping papers, and wouldn't be used for something like boxes. This is despite the fact that Scotch is a brand name of tape, and could include numerous tape products.

From Wikipedia, this is what Americans generally call "Scotch tape" (and what they would think of if they heard "Scotch tape"):

I should note that in American use, it might not always be capitalized and could very well appear as "scotch tape". I've even see "scotch-tape".