What was the main idea behind the colonists' complaint of "no taxation without representation"?

No taxation without representation means American colonist were taxed by the British Parliament and in return Americans will be given the rights to elect representatives.

Explanation:

Previously  the British Parliament had a law where the American colonist had to pay taxes but on the other hand they were not given the right to elect their representatives. The Americans complaint against it and demanded for "no taxation without representation".

The main idea behind it was that as the Americans had to pay huge taxes they should be given the right to elect their representatives as the Britishers get, that is the opportunity to elect the representatives.

europe would not recognize the confederacy as an independent nation.

i believe it was spain