The Presidency of the PRC is the head of state, and is mostly a ceremonial position with little actual power. A good analogue is the president of the State of Israel.

The Premier is the head of government, so the equivalent in a western government would be prime minister. It is however, a state post rather than a party post. In China the party supersedes the state so they are not the "leader" of the country as they would be in a western democracy. It is frequently given to the "number two" in the Chinese leadership. Examples are Zhou Enlai and Zhu Ronji.

The Paramount Leader is an informal title mostly associated with Deng Xiaoping in acknowledging his leadership of the CCP without holding the posts of either General Secretary nor Chairman who are nominal party leaders. Deng era CCP was led by a clique of party elders who formally held advisory positions. But their seniority, prestige participation in the Chinese revolution itself and the relative institutional weakness of post-cultural revolution CCP meant they had real control over the party until early-mid 1990s.