endorse
To publicly say you support or approve of something.
Endorse means to publicly support or approve of something or someone. When a famous athlete endorses a brand of sneakers, they're telling the world, “I trust this product and think you should too.” When your teacher endorses your idea for a science fair project, she's giving it her approval and backing.
Today, when you endorse a check, you sign the back to confirm you're the right person to cash it.
Politicians seek endorsements from respected leaders to convince voters they're trustworthy. A book's cover might show endorsements from famous authors praising the story. Sometimes newspapers endorse candidates, recommending who they think voters should choose.
An endorsement (the noun form) is the act of supporting something, or it can mean the statement of support itself. A celebrity's endorsement can make a product popular, while a parent's endorsement of your responsible behavior might convince another parent to let you stay over at their house.
When you endorse something, you put your reputation behind it, telling others they can trust your judgment. That's why endorsements carry weight: people know you're personally vouching for whatever you're supporting.