three-year-old
Refers to a child of three years of age. Often used in contexts like 三つ子の魂百まで (the soul of a three-year-old lasts a hundred years).
triplets
Refers to three children born from the same pregnancy. The word is the same as the 'three-year-old' sense, so context is essential.
She gave birth to triplets.
As the saying goes, 'the soul of a three-year-old lasts a hundred years,' so early discipline is important.
Standard kanji spelling for both 'three-year-old' and 'triplets'.
Rarely used kanji form; 児 explicitly indicates a child, but 三つ子 is overwhelmingly more common.
From 三つ (three) + 子 (child). The reading みつご is a regular combination of みつ and こ, with rendaku voicing of こ to ご.