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I know how conduction makes the skin warmer: if a hot metal bar is kept in contact with my skin, the high energy molecules of the bar make the molecules of my skin vibrate. That's why I feel warmer.

When photons from the sun hit my skin, do they make the electrons of the skin molecules excited and in turn increase the energy of the molecules causing them to vibrate faster?

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In principle yes, but skin contains many large and complex molecules and thus many chemical reactions can be introduced by absorbing photons. For example, UV light can even damage the DNA (just google for sunburn). Human blood cells absorb blue and green, but reflect red, giving blood its red color (in contrast, plants reflect green thus being green). In general, melanin is the most important substance in human skin interacting with visible light thus producing the color of the skin.

So different substances and different reactions, but the result is that energy is transferred to your skin.

Note, what we call infrared radiation (long wavelengths) does not introduce chemical reactions but just excite rotation and vibration modes, as you mentioned.

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