How does a vector measurement differ from a scalar measurement?
By definition, vector is a combination of a direction and value, while scalar is characterized only by value.
To measure a vector is to characterize both its direction (in one form or another) and a value (usually, a number). For example, velocity of a car moving on the road is a vector and, to measure it, we have to specify its direction (e.g. North or at angle 20 degrees from the North) and value (e.g. 40 m/sec). Another example - the force of gravity acting on a body positioned on the surface of the Earth). This force has a direction towards the center of the Earth and a value, which we call weight, measured in some units, like kilograms.
With vectors it's important to understand the space it exists in. For example, a velocity of a car on the road is a vector in two-dimensional space, while a velocity of an airplane is a vector in three-dimensional space. Depending on dimensionality, the direction is specified differently.
To measure a scalar is to specify only its value, usually, as a number. For example, speed is a scalar and, to measure it, we have to specify its value (e.g. 40 m/sec). Another example - a tire pressure measured in some units (like pounds per square inch or newtons per square meter).
Interestingly, we can consider a two-dimensional vector as a pair of scalars, if we introduce a Cartesian system of coordinates in our two-dimensional space and associate a vector with a directional segment from the origin of coordinates to some point. Then two coordinates of this endpoint fully define both direction and value of our vector. Similarly, three-dimensional vectors can be described by a triplet of scalars - coordinates of its endpoint in a three-dimensional Cartesian coordinate system.