The domain is the set of feasible input for the function. For example, functions with denominator can't have a zero denominator, logarithm can't have zero or negative inputs, and even roots can't have negative inputs.
Visually speaking, the domain is a subset of the x axis (or possibly the whole axis), so the domain gives you an idea how "how far the function goes", horizontally.
The range is the set of all possible outputs of your function, depending on all possible inputs.
Visually speaking, the range is a subset of the y axis (or possibly the whole axis), so the range gives you an idea how "how far the function goes", vertically.
