In sky surveys, what does inverse variation really mean? How do we use inverse variation to remove noise? I am trying to work on data from a .fits file and it has an IVAR header. How can I use inverse variation information along with the actual signal data to get better results or reduce noise?
It is inverse variance. A larger inverse variance means the value in that pixel is less uncertain (in a relative sense).

The inverse variance can be used as a weighting factor - it directly gives you level of significance for the measurement. It can also be set to zero for measurements that should not be used (for whatever reason).

An example would be finding the mean value of a set of pixels. A standard mean would apply equal weight to all pixels. A weighted mean would be calculated as
$$ \bar{x} = \frac{\sum_i x_i w_i}{\sum_i w_i},$$
where $w_i$ here would be the inverse variance of each pixel.