A manner is (1) a way of doing something, (2) a bearing or demeanor, and (3) a type. The plural form, manners, refers to a manner of behavior considered to be social correct.
Constructions involving manner can often be shortened to single adverbs. For example, in a calm manner and in a public manner can give way to calmly and publicly. Some such phrases lack one-word equivalents, however—e.g., in a timely manner.
A manor is (1) the estate of a European lord, or (2) the main house of a large country estate.
Examples
Manner
Second, the plan was rejected in the most democratic manner possible – through a referendum. [Guardian]
Shaunie, as Bruce calls him, has a manner that’s both friendly and immediate. [The Age]
What manner of man is this, or what manner of creature, is it in the semblance of man? [Dracula, Bram Stoker]
Manor
He didn’t do himself many favours being seen riding with his local fox hunt like a latterday lord of the manor. [Independent]
Baltic landlords, who remained overwhelmingly German, were often absent from their estates, preferring life in the towns to the isolation of their manors. [The History of the Baltic States, Kevin O’Connor]
Lerner, who lives on an adjacent 200 acres in an elegant restored cabin, says the manor house is used once or twice a month for charitable events. [Washington Post]