What is a coset? Honestly. I just don't understand the word and keep running into highly technical explanations (wikipedia, I'm looking at you!). If somebody would be so awesome as to explain the concept assuming basic knowledge of group theory and high school algebra I would be delighted.

Here's a small finite group example using $G=(\{1,-1,i,-i,j,-j,k,-k\},\cdot)$ where $i^2 =j^2 =k^2 =-1$, $-i=(-1)i,$ $1^2 =(-1)^2 =1$, $ij=-ji=k$, $jk=-kj=i$, and $ki=-ik=j$.
Having a Cayley table helps to see this structure.
$$
\begin{array}{r|rrrrrrrr}
 * &  1 & -1 & i & -i & j & -j & k & -k \\
\hline
 1 &  1 & -1 &  i & -i &  j & -j &  k & -k\\
-1 & -1 &  1 & -i &  i & -j &  j & -k &  k\\
 i &  i & -i &  1 & -1 &  k & -k & -j &  j\\
-i & -i &  i & -1 &  1 & -k &  k &  j & -j\\
 j &  j & -j & -k &  k &  1 & -1 &  i & -i\\
-j & -j &  j & k  & -k & -1 &  1 & -i &  i\\
 k &  k & -k & j  & -j & -i &  i &  1 & -1\\
-k & -k &  k & -j &  j &  i & -i & -1 &  1\\
\end{array}
$$
Clearly, $H=(\{1,-1\},\cdot)$ is a subgroup.  If I multiply all my members of $G$ with the members on the left, 
$$1\{1,-1\}=\{1,-1\},$$ $$-1\{1,-1\}=\{-1,1\}=\{1,-1\},$$ $$i\{1,-1\}=\{i,-i\},$$ $$-i\{1,-1\}=\{-i,i\}=\{i,-i\}$$ $$j\{1,-1\}=\{j,-j\}$$ $$-j\{1,-1\}=\{-j,j\}=\{j,-j\}$$ $$k\{1,-1\}=\{k,-k\}$$ $$-k\{1,-1\}=\{-k,k\}=\{k,-k\}$$ 
I can see that my cosets of my subgroup $H$ are $\{1,-1\}, \{i,-i\}, \{j,-j\}$ and $\{k,-k\}$.  If I multiply on the right ($Hg$), I think I can convince you without work that the left cosets are equal to the right cosets.  In that case, our subgroup is defined as Normal.  
Obviously these are not the only cosets I can find, but the cosets are in relation to a particular subgroup.  If my subgroup was $N=(\{1,-1,i,-i\},\cdot)$, my cosets would be $\{1,-1,i,-i\}$ and $\{j,-j,k,-k\}$. I hope this helps.  Perhaps you could try to find the left and right cosets of the subgroup $(\{1,i\},\cdot)$
If my operation were addition, my cosets would just look like $g+H$ or $H+g$.  
