a unit vector
A unit vector is a vector whose only job is to point: it has a length of exactly one (one unit) and no other significance. Think of it as a bare arrow of standard length that answers 'which way?' while carrying no 'how much'. It lets you separate a vector's direction from its size cleanly.
Any vector can be written as its magnitude times a unit vector in its direction: v = |v| times v-hat, where the little hat (as in v-hat) marks a unit vector. The standard unit vectors along the x, y and z axes are named i-hat, j-hat and k-hat, each of length 1. So a vector with components (3, 4) is written 3 i-hat + 4 j-hat. To make a unit vector pointing along any vector v, divide the vector by its own magnitude: v-hat = v / |v|, which keeps the direction but scales the length to 1.
Unit vectors are the tidy bookkeeping of vector algebra: they let you state directions unambiguously and write vector equations compactly. Whenever you see i-hat and j-hat in a physics text, they are just the 'east' and 'north' of the coordinate system, each one unit long.
The velocity 3 i-hat + 4 j-hat (m/s) has magnitude sqrt(3^2 + 4^2) = 5 m/s. Its direction as a unit vector is (3/5) i-hat + (4/5) j-hat, the same heading with the length trimmed to exactly 1.
Divide a vector by its length to keep the direction and drop the size.
A unit vector is dimensionless, pure direction. The physical units of v = |v| v-hat live entirely in the magnitude |v|, not in the hat.