Let $V$ be an affine complex variety. Let $x \in V$ be a closed point. Then a tangential base-point at $x$ is $x$ together with a regular function $t$ on $V$ that is zero exactly on $x$ (to degree $1$).
My familiarity with this notion is very meek, and I know that there are generalizations for other fields, and for normal crossing divisors (instead of one closed point; in this case they are called tangential morphisms), but I'm not sure I can recite a definition for these. In texts where I've seen this notion introduced, it said that the intuition should be that $t$ gives a direction: look at $t=\epsilon$ for $\epsilon$ going from $0$ to a small real positive number.
This hardly seems like motivation! Why would one want a base-point with a direction? What utility does it have? What problem does it solve? I've seen this in several papers, and I do wish to get over the hump and actually start understanding this.