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The term integration time is used for radio telescopes and CCDs, for the length of time over which data is collected and combined. Thus 20 seconds of integration time means an image (or even just one pixel, depending upon the configuration) is created by gathering photons for 20 seconds. Alternately, those 20 seconds could have been dedicated to gathering two images, each with 10 seconds of integration time.
It generally means the time over which an analog signal is converted into a digital count representing an integration of the signal. Such an integration time may be chosen as best to reduce the relative effects of systematic noise and/or random noise. As a general rule, lengthening the integration time improves the signal-to-noise ratio (SNR). For some objects, minimums for workable integration times can be calculated from the temperatures of the observed object and the observing instrument.
It basically means the same as exposure time (per photography). I imagine the word integration is used because mathematically, the rate of the magnitude of the signal received is integrated (per calculus) over time, the result of which is recorded.
For transients, there is a trade-off between the integration time length needed to achieve a usable SNR versus the time-resolution needed to sense the ongoing changes of interest.