I'm reading an article and it says:
Let $(\hat i, \hat j, \hat k)$ be the ball-centric coordinate reference frame illustrated in Fig. 1.
What is this "ball-centric coordinate"? I thought it was just the position $(x,y,z)$ of the ball center in the world, but I might be wrong. The variables have this hat ^ over them, that is not common to me either.
This is almost more of an English question than a math question. It's not "(ball-centric coordinate) (reference frame)", but "(ball-centric) (coordinate (reference frame))".
The adjective "ball-centric" just means something like "centered at the location of the ball". And "coordinate reference frame" is something like "reference frame for coordinates". In other words, the coordinates they will use are based on the reference frame in the diagram which is centered at the ball's location (in context: the ball's center).
The remaining question is "what does the reference frame $\left(\hat{\boldsymbol{\imath}},\hat{\boldsymbol{\jmath}},\hat{\boldsymbol{k}}\right)$ mean?" Well, in physics notation especially, the "hat" on a vector like $\hat{\mathbf{v}}$ means that the vector has, or has been made to have, unit length/magnitude. And $\hat{\boldsymbol{\imath}}$ or similar typically denotes the unit vector in the "$x$-direction": $\langle1,0,0\rangle$, etc. So that a vector $\langle1,2,3\rangle$ is equal to $1\hat{\boldsymbol{\imath}}+2\hat{\boldsymbol{\jmath}}+3\hat{\boldsymbol{k}}$.
But here we're not just talking about vectors with no fixed position, but a whole coordinate system. So $2\hat{\boldsymbol{\jmath}}$ would denote the location that is two units away from the center of the ball in the $\boldsymbol{\jmath}$ direction indicated in the picture, etc.