I have never studied algebra thoroughly, so I do not have any experience with groups and no mindset that handles them naturally. For example, yesterday I have been looking into examples of quotient groups, since that happened to be less obvious to me than quotient topology, hence the question.
Let us consider the following space $X$:
Let $A$ be the green and red circle with a bit of the blue one, and $B$ be the blue and red circle with bit of green one, so that they are both open and $C = A\cap B$ is homotopy equivalent to the red circle. Let $x_0$ be a point where green and red circles intersect. Then $\pi_1(A, x_0) = \pi_1(B,x_0) = \Bbb Z^{*2}$ and $\pi_1(C, x_0) = \Bbb Z$.
My understanding is that $\pi_1(X, x_0) = \Bbb Z^{*3}$ which can be found if we took $A$ being only the green circle with a bit of red, since their intersection with $B$ is null homotopic. But to understand interplay between quotient groups and free product of groups I want to obtain $\pi_1(X, x_0)$ from $A$ and $B$ as defined originally. I obtain that $$ \pi_1(X, x_0) = \Bbb Z^{*3} = (\Bbb Z^{*2} *\Bbb Z^{*2})/N $$ where $N \leq \Bbb Z^{*2} *\Bbb Z^{*2}$ is the normal subgroup, corresponding to $\pi_1(C, x_0)$.
What is not clear to me: what is $N$ here? I guess we can use the fact that $\pi_1(C, x_0)\leq \pi_1(A, x_0)$ and $\pi_1(C, x_0)\leq \pi_1(B, x_0)$ somehow, so letters from $\pi_1(C,x_0)$ when treated in the free product should be somehow identified? Also, what exactly is $N$ as a subgroup of the product, to which words does it correspond, how can we show those words form a normal subgroup and what are the cosets of that subgroup in the product? The latter is important for my understanding, since for now I think about elements of the quotient subgroup as cosets of the original one.
To get off the ground, it really helps to use more informative notation for $\pi_1(A)$ and $\pi_1(B)$ than just the plain vanilla notation $\mathbb Z^{*2}$.
Let's name three loops: $\gamma$ is the green loop based at $x_0$; $\rho$ is the red loop based at $x_0$; and $\beta$ is the loop based at $x_0$ which goes along the top half of the red loop, around the blue loop, and back along the top half of the red loop.
Now we can name free bases for the two rank $2$ free groups: $$\pi_1(A,x_0) \approx \langle \gamma, \rho_A \rangle $$ $$\pi_1(B,x_0) \approx \langle \rho_B, \beta \rangle $$ My reason for using the notations $\rho_A$ and $\rho_B$ is that they actually represent two different homotopy classes: the homotopy class of $\rho$ in $A$, and the homotopy class of $\rho$ in $B$.
You wrote:
Wellllllll......., that's a tad vague.
Here's a precise description: $$N < \pi_1(A,x_0) * \pi_1(B,x_0) $$ is the normal subgroup generated by $\rho_A \rho_B^{-1}$, namely the subgroup consising of all products of conjugates $w \rho_A \rho_B^{-1} w^{-1}$ where $w \in \pi_1(A,x_0) * \pi_1(B,x_0)$ is arbitrary. The intuition here is that $\rho_A \rho_B^{-1}$ represents first going around $C$ in one direction inside $A$, and then going back round $C$ in the reverse direction inside $B$. But in $X$, the "inside $A$ and $B$" doesn't matter, this loop will represent the identity element in $\pi_1(X,x_0)$.
Normality of $N$ is a simple consequence of the fact that it is generated by a conjugacy class: in any group, the subgroup generated any conjugacy class is normal.
I'm not sure what to say regarding your insistence on thinking about cosets of $N$. To me, the group theoretic idea of this quotient operation is that for any word on the letters $\gamma, \rho_A, \rho_B, \beta$, modding out by $N$ allows you to identify the two elements $\rho_A,\rho_B$ to a single element named $\rho$, thus yielding the free group with free basis $\langle \gamma,\rho,\beta\rangle$.