Consider the following sentence:
If you prefer small towns, New York City is a bad place to live.
Now imagine quoting only the part that says:
New York City is a bad place to live.
Strictly speaking, this is an accurate quote, because the quoted part is taken directly from the original sentence with no changes. But it's very misleading, because the quote omits important information that you need to understand what the speaker was really saying. That omitted information is context.
Context is the information necessary to correctly interpret a word or phrase that could otherwise be interpreted in a number of different ways. Without context, the speaker appears to be simply saying that New York is a bad place to live for everybody. When the context is supplied, it's clear that the speaker is only saying that New York is a bad choice for people who prefer small towns. That's why it's unfair to quote someone out of context--to repeat something the person says without enough information to prevent readers or listeners from misunderstanding the true meaning of the quote.
Similarly, if you ask a question about English here, you may be asked to provide additional context, such as an example sentence that illustrates the problem you want to solve. Many words and phrases can have different meanings based on context, so it may not be possible to answer the question without more information that narrows down the possible answers.