Step 1: Recall the carbylamine test.
When a primary amine is warmed with chloroform and alcoholic KOH, it makes a foul-smelling isocyanide. Only primary amines give this. \[ RNH_2 + CHCl_3 + 3KOH \rightarrow RNC + 3KCl + 3H_2O \]
Step 2: Check trimethylamine.
$(CH_3)_3N$ has three groups on nitrogen, so it is a tertiary amine. It gives no test.
Step 3: Check dimethylamine.
$(CH_3)_2NH$ has two groups on nitrogen, so it is a secondary amine. It gives no test.
Step 4: Check aniline.
$C_6H_5NH_2$ has one $-NH_2$ group, so it is a primary amine. It will give the carbylamine test.
Step 5: Check triethylamine.
$(C_2H_5)_3N$ is a tertiary amine, so it gives no test.
Step 6: Pick the primary amine.
Among all the choices only aniline is a primary amine, so only it responds to the test.
Step 7: State the final answer.
The amine that gives the carbylamine test is:
\[ \boxed{\text{Aniline}} \]