since the question is about gaining intuition of fundamental groups,i would like to explain through basic examples.for eg,say you have a unit circle and a torus and one asks you to distinguish between these two surfaces.A very obvious fact distinguishing these surfaces is that a circle has presence of one hole while torus has two holes.But,this has no meaning until we define a way to mathematically represent holes and show that it is topologically invariant.
Now,how can one mathematically represent these holes?Consider a surface with a hole in it.now,consider a point $x_0$ in it and a loop based at $x_0$ going around the hole and one not enclosing the hole as we can see in the diagrams 1 and 2 below.Now try to continuously deform this loop to the point $x_0$ in figure 1 and 2.As we can clearly see in figure 1,inability to deform the loop continuously to the point $x_0$ indicates a hole.

now,we generalize above method even more.We know that the relation of homotopy splits set of all loops based at a point,say $x_0$ into disjoint equivalence classes,where a homotopy class represents set of loops continuously deformable to one another.Moreover, we see that a loop going twice around a hole is not in the homotopy class of loop going once aroud the hole.so,we can associate integers n$\gt0$ ,n =0,n$\lt0$ to the homotopy class of loops according the no of times loop goes aound the hole in postive direction,dooes not wind around the loop and winds around the loop in negative direction.Moreover,the operation n+m has a geometrical meaning :n+m corresponds to going round the hole first n times and then m times.Thus,the set of homotopy classes is endowed with a group structure called the fundamental group which we can define as:
([f]*[g])(t) = $\begin{cases} [f(2t)], & \text{if $0\lt t \lt \frac{1}{2}$} \\ [g(2t-1)], & \text{if $\frac{1}{2} \lt t \lt 0$ } \end{cases}$
Now,we can rigorously prove that homotopy classes of loops based at a point forms group under the above defined operation (as it was intuitvely clear).Moreover,we show that fundamental groups,which gives measure of no of holes in the space is a topological invariant. Fundamental group as outlined above,detects one dimensional holes in a space.