A sufficient statistic for random variable $y$ w.r.t to r.v. $x$ is a function of $y$ that retains all of its information relevant to $x$.

An equivalent formulation is given by the Data processing theorem. See page 35 on Elements of Information theory by Thomas and Clover

A Minimal sufficient statistic is a function of every other sufficient statistic.

The notion of minimal sufficient statistics was introduced by Lehmann and Scheff´e (Lehmann and Scheff´e, 1950) as the simplest sufficient statistics, or the coarsest sufficient partition of the sample space which captures the relevant components of the sample with respect to the parameter.

An equivalent definition is that $P(y|x,T) = P(y|T)$, so that if we are given $T$, knowing $x$ doesn't change what we know about $y$. So $T$ is a sufficient statistic of $y$ w.r.t $x$.

Basically the first definition tells us that T tells us everything we need to know of y about x, and the second tells us that T tells us that knowing y or not is irrelevant for x, if we already know y, which makes sense if you think about it