To trump is to outrank or defeat someone or something, often in a highly public way. Safety might trump appearance when you're buying a car, or your desires may trump your brother's when it comes to making weekend plans.
In the card game bridge, the trump card is the most powerful card in a particular round and defeats all the others — sort of like when your needs or wishes trump someone else's. Originally trump implied a deceptive form of victory involving cheating, but that sense has been largely lost, though it's still around in the term trumped up, meaning something that's been falsely made up. A politician may face trumped up charges that could ruin his career.