Given points:
A(1, 0), B(3, 1), C(2, 2), and D(2, 1)

Step 1: Determine the lengths of the sides of the quadrilateral.
To determine the type of quadrilateral formed by these points, we first need to find the lengths of its sides.

Distance formula: The distance between two points (x1, y1) and (x2, y2) is given by the formula:
d = √((x2 - x1)^2 + (y2 - y1)^2)

Using this formula, we can find the lengths of the sides:
AB = √((3 - 1)^2 + (1 - 0)^2) = √(2^2 + 1^2) = √5
BC = √((2 - 3)^2 + (2 - 1)^2) = √((-1)^2 + 1^2) = √2
CD = √((2 - 2)^2 + (1 - 2)^2) = √(0^2 + (-1)^2) = 1
DA = √((1 - 2)^2 + (0 - 1)^2) = √((-1)^2 + (-1)^2) = √2

Step 2: Compare the lengths of the sides.
Since the lengths of all the sides are different, the quadrilateral cannot be a square or a rhombus.

Step 3: Determine if the opposite sides are parallel.
To determine if the opposite sides are parallel, we calculate the slope of each side.

Slope formula: The slope of a line passing through two points (x1, y1) and (x2, y2) is given by the formula:
m = (y2 - y1) / (x2 - x1)

Using this formula, we can find the slopes of the sides:
AB: m = (1 - 0) / (3 - 1) = 1/2
BC: m = (2 - 1) / (2 - 3) = 1/-1 = -1
CD: m = (1 - 2) / (2 - 2) = -1/0 (undefined)
DA: m = (0 - 1) / (1 - 2) = -1/-1 = 1

Since the slopes of AB and CD are equal (1/2 = 1/2) and the slopes of BC and DA are equal (-1 = -1), the opposite sides AB and CD, as well as BC and DA, are parallel.

Conclusion:
Based on the information obtained, the quadrilateral formed by the points A(1, 0), B(3, 1), C(2, 2), and D(2, 1) is a parallelogram because its opposite sides are parallel.