Vectors can have units. With common vectors like acceleration, velocity, and displacement, the unit of the whole vector is the same as each of its components. That is, a displacement vector of [3 m, 4 m, 5 m] will have a unit of meters. A force vector of [10 N, 5 N] has a unit of newtons. This can help with calculations since, for example, an acceleration vector times mass should yield a force vector.
In other areas of physics, vector components can have different units. In optical ray tracing, the components of a ray vector are a distance and an angle. So, the vector does not have an overall unit.