Step 1: Understand the set notation.
The set $k\mathbb{N}$ means all natural number multiples of $k$. For example $8\mathbb{N} = \{8, 16, 24, \dots\}$ and $12\mathbb{N} = \{12, 24, 36, \dots\}$.
Step 2: Meaning of intersection.
The intersection picks numbers that lie in both lists. A number is in both only if it is a multiple of $8$ and also a multiple of $12$, that is, a common multiple.
Step 3: Common multiples are multiples of the LCM.
Every common multiple of two numbers is a multiple of their least common multiple. So $8\mathbb{N} \cap 12\mathbb{N} = \text{LCM}(8,12)\,\mathbb{N}$.
Step 4: Find the LCM of $8$ and $12$.
Factor them: $8 = 2^3$ and $12 = 2^2 \times 3$. Take the highest power of each prime, $2^3$ and $3$.
Step 5: Multiply the prime powers.
\[ \text{LCM} = 2^3 \times 3 = 8 \times 3 = 24 \]
Step 6: Write the intersection set.
So the common elements are exactly the multiples of $24$, which is the set $24\mathbb{N}$. Therefore \[ \boxed{24\mathbb{N}} \]