Question

how do you understand the slogan no taxation without representation

Answer 1

"(No) taxation without representation" was a principle in British Law since the time of the English Civil War, which held that Parliament must levy all taxes.

This was made a legal requirement under the British Bill of Rights, passed in 1689.

The theory was that all British subjects had representation in Parliament. In practice though, in early times, direct representation of the common people in Parliament was very highly limited due to property requirements for suffrage as well as to other peculiarities of the system. 

The landed gentry effectively controlled the votes of most of the population. To circumvent arguments against the unfairness of this state of affairs, which began to arise more and more frequently after the English civil war, the doctrine of "virtual representation" was invented. It consisted, to put it bluntly, of a series of spurious arguments which implied that all British subjects actually did have representation in Parliament in spite of the plain fact that they did not.

"(No) taxation without representation" was also a phrase that became a slogan in the period before, during, and after the American Revolutionary War.

It was a brief summary of one of the primary grievances that the British Colonists in North America held against the British Crown. This was that they had no direct representation in Parliament, yet Parliament had imposed taxes on the Colonists. In North America, the Sugar Act and the Stamp Act were imposed. This, the colonists argued, was illegal under the Bill of Rights and such "taxation without representation" deprived them of their due rights as British subjects. The doctrine of "virtual representation" was almost universally rejected in the colonies.

Some went so far as to say that "taxation without representation is tyranny."

So in a sense, "taxation without representation" may be regarded as one of the major causes of the American Revolution. However, it should also be remembered that the taxes in question were not actually very burdensome, and that there were many other reasons why the Revolution happened when it did, one of those being that French power in North America had been dealt a major blow with the defeat of the French in the French-Indian War, on the Plains of Abraham, at Quebec City in 1753.

As a result the British Colonists in the original thirteen colonies that eventually revolted may have been feeling less need of the protection of the British Crown against possible invasion by the French, in the 1750's and 1760's, which is when the phrase "no taxation without representation" became a rallying cry of the revolutionaries.

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Answer 2

It was a rule in British Parliament that "Taxation without representation" so the slogan was raised before American Revolution by North American Colonies before their British Rulers to have their representation in the British Parliament. So American raised the slogan "No taxation without representation" to have legislators seats in London.
James Otis was the person who raised this issue in 1765 in the Stamp Act Congress and a small number of people supported him.