A is $3$ years younger than C but one year older than D. D is one year older than B but $4$ years younger than C. C is $15$ years old. What is the age of B in years?

Given the age of $C=15$ years.
From the statement, "A is $3$ years younger than C":

$$\text{Age of A}=15-3=12\text{ years}$$

From the statement, "A is one year older than D":

$$\text{Age of D}=\text{Age of A}-1=12-1=11\text{ years}$$

Alternatively, verifying with another statement, "D is $4$ years younger than C":

$$\text{Age of D}=15-4=11\text{ years}$$

From the statement, "D is one year older than B":

$$\text{Age of B}=\text{Age of D}-1=11-1=10\text{ years}$$

Thus, the age of B is $10$ years.