Step 1: Recall the factors governing boiling points of alcohols.
(i) Molecular mass: higher molecular mass means stronger London dispersion forces and a higher boiling point. (ii) Branching: among isomers, a more branched structure has a smaller surface area, weaker dispersion forces, and a lower boiling point.
Step 2: Group by carbon count.
Ethanol: 2 carbons; Propan-1-ol: 3 carbons; Butan-1-ol and Butan-2-ol: 4 carbons each. Therefore: ethanol $\lt$ propan-1-ol $\lt$ (both butanols).
Step 3: Compare the two butanols.
Both have 4 carbons and an $-OH$ group. Butan-1-ol is a straight-chain primary alcohol with maximum surface area. Butan-2-ol is a secondary alcohol with the $-OH$ at position 2, giving a slightly more compact structure and marginally less surface area.
Step 4: Final order.
Increasing boiling point order: Ethanol $\lt$ Propan-1-ol $\lt$ Butan-2-ol $\lt$ Butan-1-ol.