In most cases I can prove whether a relation is an equivalence relation or not but have no idea what "distinct equivalent classes" are.
I tried to read some examples but couldn't figure out how to apply them.
Would really appreciate it if you can explain what they are and how to determine them for any given relation.
Let's start with an example. You have a table with an assortment of different items. You bring in three people: somebody with good eyesight, somebody with okay eyesight, and somebody who has terrible eyesight. You ask the three people to distinguish the items on the table. The person with good eyesight sees that every object appears different, so she lists as many different items as there are objects on the table. The next person has weaker eyesight: to her, many things appear the same. When she lists the object on the table, she groups together many different objects that appear the same to her. To her, there are fewer types of things on the table. The last person, who has very poor eyesight, cannot distinguish any of the objects on the table. To her, everything is the same: there is only one kind of object on the table.
Slightly more formally, you start with a set $S$ of different elements. You put on a pair of glasses that make you less able to distinguish certain features. For example, maybe your set is the integers, and your glasses only let you see if a number is even or odd. Then you will say: "this set only has two objects". The glasses are the equivalence relation: they identify two things that are similar in a consistent way. If two things look the same, they are in the same equivalence class. If two things don't look the same, they are in distinct equivalence classes. Returning to the example of the integers and the odd-even glasses, $1$ and $2$ look different, but $2$ and $4$ look the same. So $1$ and $2$ are in distinct equivalence classes, but $2$ and $4$ are in the same equivalence class, because when I put on those glasses I can distinguish the former pair, but I can't distinguish the latter pair.
An equivalence class is the collection of all things such that $a \sim b$.
Imagine that $\sim$ means "lives in the same county as." Then, for two residents of Smith County, Alice and Bob, $\textrm{Alice} \sim \textrm{Bob}$. Eve, who lives in Franklin County, does not belong to the same equivalence class as Alice or Bob: $\textrm{Eve} \not\sim \textrm{Alice}$.
Conceptually, a map of the counties looks like a bunch of equivalence classes: counties are disjoint, as are equivalence classes. If you are in one county, you are not in another county.
For all the counties in a state, the counties partition the state. Likewise, equivalence classes partition the class on which they're defined.
Here is an example which I find useful.
Let $S$ be the set of all locations on the earth's surface and define $$X\sim Y\quad\hbox{iff}\quad \hbox{clocks at $X$ and $Y$ show the same time}\ .$$ For an example from Australia (you can supply one from your own country), we have $$\hbox{Sydney}\sim\hbox{Melbourne}\quad\hbox{but not}\quad \hbox{Sydney}\sim\hbox{Perth}\ .$$ This is an equaivalence relation, and the equivalence classes are exactly what we know as time zones.
The equivalent classes are pieces (chunks) of the set where is the equivalence relation, these pieces include the elements which are interrelated. The set of all these pieces is an example of a partition of the set where the equivalence relation is defined.
For example:
In $\Bbb{Z}$ we define $a\sim b$ if $4$ divides $a-b$, then is easy to prove that this gives an equivalence relation in $\Bbb{Z}$ and that the equivalence classes are: $$[0]=\{0,4,-4,8,-8,12,-12,...\},$$ $$[1]=\{1,5,-3,9,-7,13,-11,...\},$$ $$[2]=\{2,6,-2,10,-6,14,-10,...\},$$ $$[3]=\{3,7,-1,11,-5,15,-9,...\}.$$
You can see that these subsets (chunks) are mutually disjoint and $${\Bbb{Z}}=[0]\cup[1]\cup[2]\cup[3].$$
