As usual, we lead with a definition.
Suppose $\ltdefn{T}{U}{V}$ is a linear transformation. Then $T$ is injective if whenever $\lt{T}{\vect{x}}=\lt{T}{\vect{y}}$, then $\vect{x}=\vect{y}$.
Given an arbitrary function, it is possible for two different inputs to yield the same output (think about the function $f(x)=x^2$ and the inputs $x=3$ and $x=-3$). For an injective function, this never happens. If we have equal outputs ($\lt{T}{\vect{x}}=\lt{T}{\vect{y}}$) then we must have achieved those equal outputs by employing equal inputs ($\vect{x}=\vect{y}$). Some authors prefer the term one-to-one where we use injective, and we will sometimes refer to an injective linear transformation as an injection.