I read the following in wiki, but I can't understand what is meant by "divisor" there.
Notice that $(\mathbb{Z}/2\mathbb{Z})[T]/(T^{2}+1)$ is not a field since it admits a zero divisor $(T+1)^2=T^2+1=0$ (since we work in $\mathbb{Z}/2\mathbb{Z}$ where $2=0$)
I understand that if $a^2 \bmod p = 0$, then $p$ then isn't prime, but what did they mean about "divisor"?
An nonzero element $a$ of a ring is a zero divisor if there is another nonzero element $b$ such that $ab=0$. In particular, a zero divisor can't have an inverse since otherwise $a^{-1}ab=a^{-1}*0$ hence $b=0$, a contradiction.
That being the case, in your example $a=b=T+1$ is the zero divisor hence is noninvertible and so your ring is not a field.