To give someone the 'cold shoulder' means to ignore, or deliberately shun someone. The origin of this phrase is somewhat cloudy, but the most logical I have come across with is based around the fact that cold shoulders first published usage occured in 1816 where it shows up a Sir Walter Scott's novel, The Antiquary:
"The Countess's dislike didna gang farther at first than just showing o' the cauld shouther
It was presumably so rarely used that Scott apparently even went as far as to explain its meaning in the books glossary.
From here the phrase started to appear in many other literary works, by authors such as Charles Dickens, The Bronte Sisters, Anthony Trollope and John Galsworthy. Its sudden appearance in the 1820's suggests that Scott either invented it himself, or popularised a very uncommon phrase, and it is suggested by Micheal Quinion that:
It is much more probable that the cold shoulder was always a direct reference to that dismissive jerk of one side of the upper body to indicate a studied rejection or indifference. Scott's use of "tip the cold shoulder" and "show the cold shoulder" suggest this is so.