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If the result of a measurement is i.e. $3.2 \pm 0.7$, what is 0.7? At which confidence level we know that the real result is inside of this interval?

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As your link indicates, it is the uncertainty in the measurement. The exact meaning of this can depend on context, but most of the time it is safe to assume that this is the standard deviation in the value that you should expect if the measurement were repeated a large number of times.

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In our undergraduate lab we teach that if it's an engineering text and the confidence level is not specified, assume 95%. Although I never check that against anything other than a decades old teaching tradition – Slaviks Mar 9 '12 at 20:57
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I think it varies by field. I've heard of that 95% convention, but in experimental physics (HEP at least, since that's what I know), it's more conventional to quote the 68% confidence interval, i.e. one standard deviation. Of course the best advice is to always state the meanings of your uncertainties explicitly. – David Zaslavsky♦ Mar 9 '12 at 21:45

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