I'm struggling with finding a name for the following object:
Suppose we have a set of points. For example a triangle, including the area inside the triangle, its edges and its vertices. We then construct a sphere at each point of that triangle with the same radius for all the spheres. We then take the union of all these spheres. What would be a name for the resulting object? What would be the name for its surface?
Also, what would be the name of such a shape obtained through the same process but with a tetrahedron instead of a triangle?
As mentioned in my comment, this is a notion that is given many names in the literature. Let $S \subseteq \mathbb{R}^d$, and $B_r(x)$ denote the $d$-dimensional ball of radius $r$ centered at $x$. Then it is a cute exercise of metric set theory to prove that:
$$S + B_r(0) = \{x \in \mathbb{R}^d : \exists s \in S \text{ such that } d(s,x)\leq r\} = \bigcup_{s \in S} B_r(s)$$ Where $A + B = \{a + b : a \in A, b \in B\}$ is the Minkowksi (or set) sum of $A$ and $B$. The first set in the above chain of equalities is known either just as a Minkowski sum, or as a generalized ball (can you see which choice of $S$ recovers the standard ball?); the second set is known as an $r$-neighborhood, fattening, or thickening of $S$. Obviously, the third set is the one you described.