Current density is the amount of current divided by the cross section perpendicular to the direction of the current. A wire is basically a long cylinder that conducts current. A cross section of a wire is circle. So the current density of a wire is the current divided by the cross section area.
J = I/A
By the way, I find it helpful to check the units of a quantity to better understand it. Current Density J has units Amps/m2.
Yes. The definition is right but the unit normally taken is amperes per square cms. Usually for all enclosed busbars its safe and economical value is between 96 & 128 A/sq.cm.