Semantics

arm
К утру у меня онемела вся левая рука — от плеча и до кончиков пальцев. In the morning, my whole left arm was numb, from my shoulder to the tips of my fingers.
hand
Он протянул мне руку. He extended his hand to me.
Russian differs from English in that the word рука may denote either "hand" (i.e. that part of the body from the wrist to the tips of the fingers) or "arm" (i.e. that part of the body from the shoulder to the tips of the fingers). This entry is concerned with both of these usages. The word рука is not normally used to refer to animals' arms or hands.

For distinctions between the word кисть "hand" and its synonym рука "hand," see Lexical Relationships: Synonyms.

The word рука also has a number of other metonymic and metaphorical meanings: a style of drawing, painting, or writing, e.g.:
Посмотрев на письмо, он узнал руку брата. He looked at the letter and recognized his brother's hand.
a creative style, e.g.:
Здесь видна рука большого мастера. The hand of a great master is in evidence here.
a characteristic method, e.g.:
В этом заговоре видна была рука КГБ. The hand of the KGB was evident in the plot.
manual laborers, e.g.:
Стране нужны рабочие руки. The country needs blue-collar workers (lit. "laborers' hands").
a person who has possession of something, e.g.:
Эта картина находится в частных руках. This picture is privately owned (lit. "in private hands").
a proposal of marriage, e.g.:
Он просил её руки. He asked for her hand.
a direction or side, e.g.:
Он уселся на скамье по правую от неё руку. He took a seat on the bench on her right hand side.