How is "smallest" defined with regards to the partial order $\subset$? If we define a partial order $\preceq _R$ on any set $A$ by $B\preceq_R C$ if and only if $B\subset C$ for subsets $B$ and $C$ of $A$. Then how would the term "smallest" be interpreted in this relation?
I ask this, because when we define the transitive closure of a relation we define it by saying that for a relation $R$ on a set $S$, the transitive closure of $R$ is the "smallest" set that contains $R$ and is transitive. I have been told that "smallest" here refers with respect to the partial order relation $\preceq_R$ defined at the beginning.
Could someone give an example of this and maybe explain how it should be interpreted. I am assuming we say $B$ is smaller than $C$ if $B\preceq_R C$. Is this the case?
 A: Yes, smaller and bigger are exactly the intuitive definitions in a partial order. Formally, those concepts are kind of interchangeable, but given a specific order and a convention of what should be thinked as "bigger", it's just the natural idea. But this does not address what "smallest" means and your specific case, so I'll explore it:

First of all, a general definition: Let $\le$ be a partial order on some set $S$. An element $a \in S$ is said to be minimal (resp. maximal) if $\forall\ b:\ b \le a \implies b = a$ (resp. $a \le b \implies a = b$); that is, it's pretty much the intuitive notion of minimum (but those are actually not the same!). A minimal element does not need to be comparable with every other term. When it is comparable, we call it a minimum (and, informally, the "smallest"), and it's not hard to prove that it is unique.
Specifically, when working with posets by inclusion,we have $S \subseteq \mathcal P(X)$ for some set $X$ and the smallest element that has a property is simply a minimum with the desired property (that might, or might not exist). In our case, the transitive closure (which you defined as the smallest transitive relation that contains $R$) does indeed exist. Fix your $R\subseteq A\times A$ and define $$S = \{\text{transitive binary relations on A that contain R}\}.$$ This is a poset with inclusion and obviously $A\times A\in S$. Now we can do this little trick: take $\bar R = \bigcap\limits_{R_S \in S} R_S$. This is a minimal element on $S$! In fact, $\bar R \ne \emptyset$ because $S$ has at least one element, and $R \subseteq \bar R$ (because each $R_S \supseteq R$); you can prove as an exercise that $\bar R$ is, in fact, a transitive relation. It is minimal because given any $R' \in S$, if $R' \subseteq \bar R$, then by construction $R'$ appears in the intersection thus $\bar R \subseteq R'$ (also, this proves that $R'$ is comparable w.r.t. every other term), so $\bar R = R'$. That is your transitive closure, the minimum on $S$.

Note that this is easier to comparatively describe/give an "universal property" than to explicitly compute.
In general, every time you have structures given by subsets/supsets of a given set, the smallest/biggest structures s.t. something is satisfied are given by this exact construction. You build the poset of all those that satisfy your condition and join/intersect all of them.
A few examples include: in topology the closure, interior; in analysis, the generated $\sigma$-algebra; in algebra, the field subextension generated by an element).
You can find a few (not so clear) examples on Wikipedia's page on minimal and maximal elements.
