What does it mean to “condition on X”?

Could someone explain what this lingo means in regular English? Example sentences for context: "The single-partition hot deck based on a metric that conditions on $$X$$ has the problem that it fails to preserve associations between observed and imputed components of $$Y$$ for each pattern"

"Another approach that preserves associations between $$p$$ variables, which we refer to as the $$p$$-partition hot deck, is the create the donor pool for $$Y_j$$ using adjustment cells (or more generally, a metric) that conditions on $$X$$ and $$(Y_1, \dotsc, Y_{j-1})$$, for $$j=2, \dotsc, p$$, using the recipient's previously imputed values of $$(Y_1,\dotsc, Y_{j-1})$$, when matching donors to recipients".

For instance, you have data on students X from a particular elementary school, maybe their ages, gender, parents' income etc. You're trying to explain their grades Y. So, you build a regression model $Y=X\beta+\varepsilon$, then use $\beta$ and their variances $\sigma^2_\beta$ to test hypotheses, maybe the impact of parents' income on grades or such.
So, in this example $\beta,\sigma^2_\beta$ and your hypothesis testing results are conditioned on X. If you plug a different sample $X'$, maybe from a different school in a different country, everything may change, your conclusions may change.