I am reading some exercises that talk about taking a loop in the fundamentla group that generates the whole fundamental group.
But I can not find the definition of generator of fundamental group in my notes, and also I can not find anything in internet. Can someone give a quick explanation about the definition and some intuition behind this concept?
In general, it can happen that a given group $G$ is cyclic. That means that there exists an element $g \in G$ so that every element of $G$ is given as a multiple (under the group operation) of $g$. Up to isomorphism the cyclic groups are exactly the quotient groups of $\mathbb{Z}$ (including the full group and the trivial group).
For arbitrary pointed spaces (if you want you can also assume path-connectedness to make the fundamental group independent of the choice of the basepoint) the fundamental group is not cyclic though and therefore you will not always be able to find/choose a generator. One conrete example where this fails would be the torus $T^2 = S^1 \times S^1$ with fundamental group $\mathbb{Z}^2$.
Per definition the fundamental group consists of loops up to homotopy and therefore a possible generator will also be a loop (up to homotopy), but as I stated before you will not necessarily find a generator. The intuition you should have would then be the following. Is every loop in your space a multiple of some specific loop (up to continuous deformations). This is true for the circle $S^1$. Every homotopy class of loops here is given by walking around the circle some amount of times (in either clockwise or counterclockwise direction which corresponds to the positive and negative integers).