Given a topological space $X$ and a point $b$ of it, what does "the isomorphism type of all $\pi_1(X,b)$" which is denoted by $\pi_1(X)$ mean? After that, I want to understand what does it mean to have trivial $\pi_1(X)$.
Do we classify fundamental groups based at different points of the space up to isomorphisms, somehow?