The shadow health minister was embroiled in a scandal after saying on the microblogging site that ''white people love playing 'divide & rule'".
The Labour MP has said that she was not generalising about race with the comments, which were made in response to criticism of media use of the phrase "black community leaders" after the Stephen Lawrence murder trial.
Abbott, the first black woman to be elected as an MP, said in a statement: "I apologise for any offence caused.
"I understand people have interpreted my comments as making generalisations about white people. I do not believe in doing that."
Labour party leader Ed Miliband has told Abbott that her remarks were "unacceptable".
Speaking to the BBC, shadow business secretary Chuka Umunna said that the case shows the "perils" of politicians using Twitter.
"Ed Miliband has spoken to her this morning and made it very clear in no uncertain terms that the contents of the tweet were unacceptable," he said.
"If Diane believed the words as they were expressed and she had not apologised then Ed Miliband would obviously have taken the requisite action.
"For us as politicians, Twitter is a very useful tool to communicate with people, but it has its perils."
Earlier today, Abbott claimed that her remark had been misinterpreted, telling her Twitter followers: "Tweet taken out of context. Refers to nature of 19th century European colonialism. Bit much to get into 140 characters."
In a statement, the Labour Party said: "We disagree with Diane's tweet. It is wrong to make sweeping generalisations about any race, creed, or culture.
"The Labour Party has always campaigned against such behaviour - and so has Diane Abbott."
Former Labour MP George Galloway earlier dismissed the criticism of Abbott, tweeting: "Diane Abbott has been my friend for 25 years; only the obtuse would think her a 'racist'."