What is the relationship between race and intelligence ?

Unproven theories about a link between race and intelligence are often advanced by people who are looking for ways to justify racial discrimination.
"Race" as a factor related to human brain capacity was a cornerstone of evolutionary theories of the nineteenth century, and as such, it was widely expressed in social Darwinism, a racist theory that supported the practice of colonialism. The notion that nonwhite people needed a firm guiding hand to help them overcome inferiority was used to justify European colonialist expansion.
In the twentieth century, some revivals of this ostensible correlation surfaced, most notably in the 1994 book The Bell Curve: Intelligence and Class Structure in American Life by Richard Hernstein and Charles Murray. Their claims that blacks are naturally less intelligent (which they supported in part by citing IQ tests) and that this is a reason they have had less socioeconomic success have been further refuted since then.
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The relationship between race and intelligence is complicated by numerous social factors, especially racism.
While there are no identifiable genetic differences between different human races, it cannot be denied that race is a social reality. Through its social reality, race also becomes a biological reality. The way human beings interact with one another often has to do with another person's "race," whether this is conscious or not. There are no biological characteristics we can accurately compile to create a profile of a certain "race," including phenotypic variations like hair texture and eye colour. Despite this fact, people treat each other as though they did belong to different races with different characteristics. When they decide that one race is inferior, it can affect those people's access to things like healthy food, medical care and education.
For example, black people in the U.S.A. have higher rates of diabetes than white people. This is not because black people are inertly more likely to develop diabetes. It is because black people tend to be poorer and thus not be able to access the healthy lifestyle that prevents diabetes, and the medicine that treats it. The reason for this is institutional racism, not genetic differences in black people. Nonetheless, it does prove that race can lead to differences in health. Thus, race is a biological reality.
While there are no racial predictors of intelligence (though disciplines like Anthropology have tried to say there were through outdated pseudosciences like phrenology), it certainly is possible that through the mechanisms of institutional racism that make race a biological reality, some people have less access to education. While education does not equal intelligence, and no race actually is more or less intelligent than another, it is something you should consider when answering this question.

Biologically, there is no relationship.
The suggestion that there may be a connection between race and intelligence stems from the ideological beliefs that were necessary to spread in order to keep white supremacy intact. Studies have repeatedly been unable to connect race with any significant biological difference (insignificant meaning the differences in skin pigment and hair texture).
Perhaps the perpetuation of this belief comes from the continued oppression of non-white communities in white dominated countries like the United States. The government and other power structures have historically denied people of color access to things such as education and other means of gathering knowledge like the Internet and libraries. As a result, people of color have not been able to benefit from the education that white people have been privileged enough to access.
Another structure that is of note is the actual make-up of intelligence tests themselves. Despite the fact that the creator of IQ tests himself has said that they are a poor measure of intelligence, these tests are still widely used and provide ammo for the myth that race is connected to intelligence. These tests were used to justify the eugenics movement (and subsequent sterilization of women of color) and prevent people of color from working certain jobs. This is due to something called "cultural specificity" which suggests that IQ tests are biased towards the culture that creates them- in this case, white, middle class men.
As a result, though there may be studies and tests that appear to support the idea that race is connected to intelligence, they are actually relying on a history of racism that denies the ways that oppression works to stifle the education and academic testing of people of color.

Given that race is largely a social construct rather than a biological fact, it is difficult to claim any relationship between "race" and intelligence. For example, in the United States we've inherited racial definitions that classify anyone with "one drop" of black blood as "black"—or less extremely, a legacy that defined has people with, say, one black grandparent as "black," even though they are, strictly speaking, 75 percent white. To intelligence test such a person as "black" and draw conclusions based on that premise simply doesn't make logical sense. Furthermore, DNA testing has shown that the human race is highly similar in terms of DNA, and that two people deemed "white"—say, an Italian and a Swede—can have more genetic variation than a black African and a white Frenchman. Finally, the genetic variants that can be found between blacks and whites generally cluster around phenotypical physical features, such as skin pigmentation and hair texture, that have no bearing on intelligence.
Furthermore, it should be pointed out that much intelligence testing is also socially constructed, privileging the kind of cognitive (as opposed to emotional, artistic, kinesthetic, or social) training that higher-caste groups have greater access to in Western culture.
With race as a suspect category, it is difficult to defend studies that purport to connect race and intelligence. If we could find statistically significant (i.e., large enough) groups of people of different races with verifiably very distinct genetic differences and manage to test their intelligence in culturally non-biased ways, we might learn something, but generally such studies have been sloppily done and motivated by political agendas that need to be carefully questioned, given the ugly legacy of racism in our culture and other cultures.