To push through or carry out something despite opposition; to force one's way past resistance.
彼は反対意見を押し切って新しい事業を始めた。
He started the new business despite opposing opinions.
親の反対を押し切って家を出た。
I left home against my parents' wishes.
会議での異論を押し切って決定を下した。
They made the decision, overriding objections in the meeting.
To press and cut — to cut something while applying pressure (a more literal/physical action).
野菜を包丁で押し切るようにして薄く切った。
I sliced the vegetables thinly by pressing and cutting with a knife.
和裁では布を押し切る技法が使われることがある。
In traditional sewing, a technique of pressing while cutting fabric is sometimes used.