Can someone explain the concept of ancillary statistics in layman's terms? I'm having a hard time trying to relate or understand it in the simplest way (without solving).
"Without solving" in a sense that I don't have to solve for the marginal distribution of T2, if for example there are T1 and T2, and see that it is independent of the parameter. I'm just trying to find a way how to explain it to other people in layman's terms without using any technical statistical terms
 A: The way I think of ancillary statistics is: a data set has numerous sources of randomness. A sufficient statistic carries all the information that I can extract in the data about this randomness. A minimally sufficient statistic is the "smallest" sufficient statistic. Now say that this has the form $S = (T,C)$.
Now I happen to be interested only in some aspects of the randomness ($T$) in the data, and I would rather condition on the uninteresting parts (treat $C$ as actually fixed to $c$). This makes sense to do when $C$ depends only on parameters that I am not interested in, and it is especially useful when $T|C$ has a simple distribution. 
For example, I carry out a linear regression. I am interested in how some covariate affects the outcome, and I assume that $Y_i \sim N(\alpha + \beta x_i, \sigma^2)$ for $1 \leq i \leq m$.  What I care about is $\alpha, \beta$ and $\sigma^2$. 
But there are also other sources of randomness: $x$ is probably realization of a random variable $X$ (uninteresting). The sample size is probably random as well (if it is not fixed in advance), and $m$ is a realization of a random variable $M$ (uninteresting). When I carry out my linear regression I condition on these aspects of the data. One reason is that I am not really interested in $X$ and $M$ per se. The second reason is that it would be very complicated to write a likelihood for all sources of randomness $(Y_i, X_i, M)$, $1\leq i\leq M$ at the same time. 
If there is a minimal sufficient statistic $S=(T, C)$, and $C$ describes exclusively $X$'s and $M$ then I would rather condition on $C=c$, which is the same as treating $X_i = x_i$ and $M = m$ as fixed. 
