What are the circumstances of acquisition for the United States?

This question is not exactly clear.  I am going to assume
that you are asking how the land territory of the United States was
acquired.

The first part of the United States was acquired
by winning the Revolutionary War.  After that war, the United States was made up of
everything from the Great Lakes down to the Gulf of Mexico, east of the Mississippi
River (except for Florida).

The next big acquisition was
the Louisiana Purchase of 1803 when the US purchased huge areas of land from France. 
After that, the US got much more land by defeating Mexico in a war that lasted from 1846
to 1848.  This gave the US most of the Southwest.

There
were other, smaller acquisitions through treaties and purchases (Florida, the Oregon
Territory, Alaska) or through conquest (Hawaii).

Overall,
then, most of the US was acquired by treaty and by buying.  The major exception to this
is the Mexican Cessian, which was taken by force.