Why do people call Morgan Freeman's character Red?

Well, it is kinda explained on The Daily Beast,

In the book, Ellis Boyd "Red" Redding was a ginger-haired, middle-aged Mick. Harrison Ford, Clint Eastwood and Paul Newman were all considered for the role that went to Morgan Freeman. Darabont alluded to the unusual casting choice by having Red jokingly reply to Andy’s inquiry about his nickname with the line, "Maybe it's because I'm Irish." Happily, they opted to not follow the quote with audio of a studio audience laughing.

Anyway, it was derived from his actual surname of Ellis Boyd Redding, or 'Red' for short, as revealed in his first parole hearing in the film.

Borrowed from comments:

The physical characteristic of red hair is most common in the Irish. And for non-native speakers, "ginger-haired" means "red-haired".

Credits to @G&C, "red-haired" is used to describe orange hair colours because orange is (relatively speaking) the newest colour name in the 7-colour spectrum (besides indigo), so the term "red-hair" literally precedes the use of "orange" as a colour.

There's more to the answer. Yes, it's because Irish people are known to have red hair (although this is actually from the Norse influence).

But it also a joke about race, and his accent or lack thereof. It's absurd that he could be Irish since blacks are not natives of Ireland (and still only constitute 1.4% of the current population) and that he has no discernible Irish accent.

As previous answers have said, his name is "Redding".

Interesting though, "red" or "redbone" is an African-American skin/hair tone where the person has a multi-racial heritage giving them a very light skin with a slight reddish tint, as opposed to "yellow" or "high yellow" with a more yellow tint. The politics of these phrases are beyond complicated, as with anything to do with African-American racial-heritage issues. Shawshank Redemption starts in the era of the KKK and Jim Crow though, so political correctness was very much not a thing back then. Morgan Freeman's skin is darker than would typically be described as "red", but it's worth noting that this is actually a legitimate nickname for an African-American character.

"Red" was short for his family name "Redding". It is quite possible that he had Irish ancestry, and an Irish family name.

There was quite a history of intermarriage and intermixture between Irish immigrants to the US and African-Americans in the 18th, 19th centuries.

NYT: How Green Was My Surname; Via Ireland, a Chapter in the Story of Black America

So many African-Americans have Irish-sounding last names that you would think that the long story of blacks and Irish coming together would be well documented. You would be wrong.

... when it comes to written historical exploration of black-Irish sexual encounters, "there are little mentions, but not much."

... unlike native-born whites, "Irish were more willing to accept and acknowledge interracial allegiances.''

Before the Civil War, "the free mulatto population had the same number of black moms as white moms."

... mixed-race children would have been given Irish surnames when their Irish fathers married their black mothers, or when their unmarried Irish mothers named children after themselves.

The Irish ended up in the Caribbean, too. Britain sent hundreds of Irish people to penal colonies in the West Indies in the mid-1600's, and more went over as indentured servants.