To hold something in one's mouth (e.g. a stick, food); often written in kana rather than kanji.
犬がボールをくわえて庭を走り回っている。
The dog is running around the yard with a ball in its mouth.
赤ん坊はよくおしゃぶりをくわえている。
Babies often have a pacifier in their mouths.
彼は虫をくわえて見せた（＝口に含んで見せた）。
He put an insect in his mouth and showed it (i.e., held it in his mouth to show).
To bring along or include something/someone; a broader sense of taking or adding something with you.
釣りに行くときは、餌をいくつかポケットにくわえて行ったほうがいい。
When you go fishing, it's good to take some bait in your pocket.
彼はいつも小さなノートをカバンにくわえている。
He always carries a small notebook in his bag.