You are given a set $A = \{1,3,5,9,11,18\}$ and a relation $R$ on $A$ given by
$$R = \{(a,b) \mid a - b \text{ is divisible by } 4\}.$$
For example, $(1,9) \in R$ because $1 - 9 = 8$ is divisible by $4$.
Now, any time we have an equivalence relation on a set, this relation partitions the set into disjoint sets called equivalence classes, where each equivalence class is given by those elements that are equivalent to each other with respect to the equivalence relation. In order for this to make sense, however, we first need to check that $R$ is indeed an equivalence relation, otherwise we can not speak of its equivalence classes.
Clearly $R$ is reflexive because $a - a = 0$, and $0$ is always divisible by $4$. $R$ is also symmetric because if $a$ is divisible by $4$, then $-a$ is divisible by $4$. If $a - b$ is divisible by $4$ and $b - c$ is divisible by $4$, then $a - c = (a - b) + (b - c)$ is also divisible by $4$, so $R$ is also transitive, and hence an equivalence relation.
The equivalence classes for $R$ on $A$ turn out to be
$$C_1 = \{1,5,9\}$$
$$C_2 = \{3,11\}$$
$$C_3 = \{18\}$$
Note that these sets are disjoint, and together they contain all elements of $A$, just like we expected. To see why for example $C_1$ is an equivalence class, notice that $1 - 5 = 4$ and $1 - 9 = 8$ are divisible by $4$, so $1$ is equivalent to $5$ and $9$ with respect to $R$. However, $1$ is not equivalent to for example $3$, because $1 - 3 = 2$ is not divisible by $4$. Hence $1$ and $3$ must be in different equivalence classes.
Traditonally, we first define a partial order (sometimes even a preorder before that) to be reflexive, antisymmmetric ($xRy$ and $yRx$ implies $x=y$, this a pre-order need not be) and transitive. Lots of examples of this: order on $\Bbb Q$ adn $\Bbb Z$, inclusion relations etc. etc.
Then among those we single out linear orders (the orders we grow up with on "numbers" are mostly linear): we want comparability: $x=y$ or otherwise $xRy$ or $yRx$ must hold (not both as this would imply $x=y$ again). But then peope wanted to axiomatise $x < y$ instead of $x \le y$ and the distinguishing things there are: no reflexivity (never $x< x$ so $xRx$ is always forbidden) and asymmetry: if $xRy$ then never $yRx$), so the relation always has to decide on one of them.
A connex relation is sort of a generalisation of the linear order: we always want either $xRy$ or $yRx$. We then don't demand transivity or asymmetry or antisymmetry necessarily. So it can be a very different thing. E.g. After a round-robin tournament in a game without draws, the players will have constructed a connex relation at the end ($xRy$ : $x$ has beaten $y$), which is the also nonreflexive as you cannot beat yourself.
Linear orders I can see the importance of, and they have a rich theory (order types etc) but connex relations don't add much in my opinion. I've never seen them in any of the papers I studied.
Best Answer
The difference in definitions is that Wikipedia defines non-strict total orders (things like $x\le y$ where elements relate to themselves) and Munkres defines strict total orders (things like $x<y$ where elements do not relate to themselves). Wikipedia's connexity condition states that for any $x,y$ either $x\le y$ or $y\le x$. This is not true for strict total orders in the case where $x=y$, so Munkres' definition instead states for any distinct $x,y$ either $x<y$ or $y<x$. Annoyingly "total order" as used in math is ambiguous, we precede it by "strict" or "non-strict" if we want to distinguish which we are talking about.
For what it's worth "connex" is quite rare, I don't think I've ran into it outside Wikipedia. According to this talk page it was introduced because the previous language (calling this a "total" relation) was ambiguous.