The Coalition combat troops consisted almost entirely of US and UK forces. However, throughout the operation, rather than refer to combat forces' actual composition (mostly US, some UK and a very few Polish troops) the White House used this euphemistic term that obscured the reality that the vast majority of the troops were Americans. See the Coalition of the willing article for the specific issues and rhetoric surrounding the Iraq invasion and the use of the term The Coalition in the context of that one operation.
This obscuring label is so useful that some consider it very likely to be used in future, i.e. that the United States would never actually engage in any questionable military maneuver "alone", but would always recruit at least one tiny country, and so justify its actions in the name of The Coalition.
It is widely believed that the use of Coalition labels and rhetoric are very deliberate, masking a policy of unilateralism to appear as multilateralism. The reality is that this Coalition is nothing like the Allies of World War II, from which the basic rhetoric is drawn: in that war large nations like the USSR and China, a much larger British Empire and a great many troops from captive nations like France and Poland participated. There was a genuine multilateral coalition of many countries:
However, even one tiny country, sending no troops and being heavily funded to let its name and flag be used, plus the United States, would constitute The Coalition for purposes of any military matter whatsoever. The concept of an alliance or coalition would be simply for show, almost all troops then American.
See also War on terrorism.