How do I wait for a promise to finish before returning the variable of a function?

Instead of returning a resultsArray you return a promise for a results array and then then that on the call site - this has the added benefit of the caller knowing the function is performing asynchronous I/O. Coding concurrency in JavaScript is based on that - you might want to read this question to get a broader idea:

function resultsByName(name)
{   
    var Card = Parse.Object.extend("Card");
    var query = new Parse.Query(Card);
    query.equalTo("name", name.toString());

    var resultsArray = [];

    return query.find({});                           

}

// later
resultsByName("Some Name").then(function(results){
    // access results here by chaining to the returned promise
});

You can see more examples of using parse promises with queries in Parse's own blog post about it.

What do I need to do to make this function wait for the result of the promise?

Use async/await (NOT Part of ECMA6, but available for Chrome, Edge, Firefox and Safari since end of 2017, see canIuse)
MDN

    async function waitForPromise() {
        // let result = await any Promise, like:
        let result = await Promise.resolve('this is a sample promise');
    }

Added due to comment: An async function always returns a Promise, and in TypeScript it would look like:

    async function waitForPromise(): Promise<string> {
        // let result = await any Promise, like:
        let result = await Promise.resolve('this is a sample promise');
    }

You don't want to make the function wait, because JavaScript is intended to be non-blocking. Rather return the promise at the end of the function, then the calling function can use the promise to get the server response.

var promise = query.find(); 
return promise; 

//Or return query.find();