In an adaptive sampling design, the sampling procedure may be altered during the survey based on the values of the variable of interest that has been observed so far. Assume that the variable of interest is $y$. Also assume that the desired (fixed) sample size is $n_s$ and that you have made $m<n_s$ observations of your variable, $y_1,\ldots,y_m$. Then, in an adaptive sampling design, you would be able to change your selection procedure during the survey depending on the observed $y_1,\ldots,y_m$. So, the "values of the variable" are basically the values of your variable of interest that you have observed until the current time in your survey.
A classical example is when a rare, spatially clustered animal population is surveyed. When the researcher discovers an area where the density of animals is high (during the sampling process), he or she might want to change the sampling procedure to include only those geographical sites that are neighbors to the area with high population density, hence adapting the sampling procedure to what has been observed so far.