Step 1: Concept
According to the Lewis theory, a base is an electron-pair donor.
Step 2: Meaning
Amines consist of a nitrogen atom with three bonds and one lone pair of electrons.
Step 3: Analysis
The lone pair on the nitrogen atom can be donated to a proton ($H^+$) or a Lewis acid. This ability to donate an electron pair makes the molecule basic (alkaline).
Step 4: Conclusion
Therefore, the lone pair of electrons is the structural feature responsible for the basicity of amines.
Final Answer: (C)