The difference between $\mathbb{Z}\times\mathbb{Z}$ and $\mathbb{Z}*\mathbb{Z}$? I have some questions regarding some notations being used here.
I am still relatively new to algebraic topology, so I am a bit confused.
I saw that $\pi_1(S^1\times S^1)\simeq\mathbb{Z}\times\mathbb{Z}$ and $\pi_1(S^1\vee S^1)\simeq\mathbb{Z}*\mathbb{Z}$.
I know the difference between $\times$ and $\vee$. But what I am unsure is the difference between $\mathbb{Z}\times\mathbb{Z}$ and $\mathbb{Z}*\mathbb{Z}$.
$\mathbb{Z}\times\mathbb{Z}$ is the product group am I right? But what is $\mathbb{Z}*\mathbb{Z}$? How do we call it? I could not search since I don't know the name.
Could somebody please give some help? Thanks.

The notation $*$ denotes that you are considering the free product of the two groups. The free product of two groups is basically words in an alphabet given by them. However, we also require that each word is fully reduced (i.e $aa^{-1} = e, aa = a^2$) in the dictionary of words that can be described by the alphabet. 
