In abstract algebra (modern algebra), is the equivalence classes the same thing as cosets? In the lecture notes that I have, it seems as though they are but is it a universal rule for the equivalence classes to mean the same thing cosets?
or is the equivalence classes a subset of some set that equivalence relation holds ie (reflexivity, symmetric and transitivity)?
As Hagen von Eitzen's answer shows, every coset is an equivalence class of a particular equivalence relation: if $H$ is a subgroup of $G$, this induces an equivalence relation $\sim_H$ given by $a\sim_Hb\iff ab^{-1}\in H$ (note that since $H$ is a subgroup, $ab^{-1}\in H$ iff $ba^{-1}\in H$, so this is in fact symmetric).
In fact, more is true: in a precise sense, every "nice" equivalence relation on a group comes from a coset! Specifically, say that an equivalence relation $\sim$ on a group $G$ is a congruence if
for all $a, b, c\in G$, $a\sim b\iff a^{-1}\sim b^{-1}$, and
for all $a,b, c, d\in G$, if $a\sim b$ and $c\sim d$ then $ac\sim bd$.
That is, a congruence is an equivalence relation that respects the group structure. Then for a congruence $\sim$, let $H_\sim=\{g\in G: g\sim e\}$ be the $\sim$-class of the identity of $G$; it's not hard to show that $H_\sim$ is a subgroup, and the $\sim$-classes are exactly the $H_\sim$-cosets.
That said, there are lots of "bad" equivalence relations on a group. For instance, on the group $(\mathbb{Z}, +)$ let $\sim$ be the equivalence relation given by $$m\sim n\iff m=n=17\mbox{ or }m\not=17\not=n;$$ this divides the group into two classes, $\{17\}$ and $\{$everything but $17\}$. This equivalence relation of course does not come from a coset, but is an equivalence relation nonetheless.
OK, that's a stupid example. But there are even interesting equivalence relations on groups that aren't congruences, so don't divide the group into cosets of some fixed subgroup. For example, consider the relation $g\sim h$ if there is some group automorphism $\alpha$ of the group with $\alpha(g)=h$. This is an equivalence relation on $G$ (automorphisms are invertible and composable) whose classes are called orbits; it's studied seriously (I've actually seen it most in mathematical logic!) but in general is not a congruence.
That said, based on personal experience I'll (tentatively) say that most interesting equivalence relations on groups are congruences.
They are related.
For example if $G$ is a group and $H$ a subgroup, then cosets are subsets of $G$ that are of the form $aH$ with $a\in G$. By the group properties of $H$, we can infer that cosets are either disjoint or identical. Also, they are all non-empty and their union is all of $G$. That makes the set of cosets a partition of $G$. And a partition is "the same" a an equivalence relation, a la $x\sim y$ iff $x$ and $y$ are in the same part of the partition (and the parts of the partition are then jsut the equivalence classes of the equivalence relation $\sim$).
But of course we could define totally different equivalence relations (or partitions) on $G$ where the equivalence classes are not cosets.