Definition
A family of random variables $(X_n)_{n\ge 0}$ is uniformly integrable if
$$ \lim_{K\to\infty}\sup_{n\ge 0} \mathbb E\bigl[|X_n|\mathbf 1(|X_n|\ge K)\bigr]=0. $$Interpretation
Uniform integrability means the tails of the whole family are controlled uniformly in $n$.
Common trap
It is not enough that each fixed $X_n$ has a small tail for some threshold depending on $n$. UI requires one large cutoff $K$ that works simultaneously for all $n$.