Well a tensor roughly speaking is a function that can take a vector as an input. A tensor also can take a "one-form" as input too, but that's a bit more complicated. Moment of inertia tensor is a such a function.
For an object that is not symmetrical in 3d will have different moments of inertia in x, y and z directions. A moment of inertial tensor is usually represented as a 3d matrix containing all the components of moments of inertia.