A formal name for "smallest" and "largest" partition
I do not think that there is terminology, which would be universally agreed. However, Google search suggests that some people use the following names: trivial partition for the partition $\{A\}$ and discrete partition or singleton partition for the partition the partition $\{\{a\};a\in A\}$.
Interestingly enough ProofWiki Article suggests that trivial partition is used for both of them and calls the partition $\{A\}$ singleton partition, although no reference is given there. The name used for $\{\{a\}; a\in A\}$ at ProofWiki is partition of singletons.
Some examples of the use of the names mentioned above:
Introduction to Abstract Algebra By W. Keith Nicholson uses the name trivial partition and singleton partition, see p.19
An Introduction to Measure Theory By Terence Tao uses the name discrete partition, see p.68
Google Books:
- trivial partition
- discrete partition
- singleton partition
- partition of singletons
Suppose that $A$ is a non-empty set and $P_1,P_2$ are two partitions of $A$. We say that $P_1$ refines $P_2$ if for every $B\in P_1$ there is $C\in P_2$ such that $B\subseteq C$. This means that $P_1$ was a result of partitioning each part of $P_2$.
We also say that $P_1$ is finer than $P_2$; that $P_1$ is a refinement of $P_2$; or that $P_2$ is coarser than $P_1$.
It is a nice exercise to verify that the relation $x\prec y\iff x\text{ refines }y$ is a partial order over the partitions of $A$, in fact it is a lattice. Every two partitions has a least-refinement and a maximal-coarse partitions.
One can now see that the partition to singletons is the maximum of this partial order. Indeed it is the finest partition. Similarly the partition into a single part is the coarsest partition, and it is the minimum of the order.