Solution that works in all modern browsers:

function contains(arr, obj) {
  const stringifiedObj = JSON.stringify(obj); // Cache our object to not call `JSON.stringify` on every iteration
  return arr.some(item => JSON.stringify(item) === stringifiedObj);
}

Usage:

contains([{a: 1}, {a: 2}], {a: 1}); // true

IE6+ solution:

function contains(arr, obj) {
  var stringifiedObj = JSON.stringify(obj)
  return arr.some(function (item) {
    return JSON.stringify(item) === stringifiedObj;
  });
}

// .some polyfill, not needed for IE9+
if (!('some' in Array.prototype)) {
  Array.prototype.some = function (tester, that /*opt*/) {
    for (var i = 0, n = this.length; i < n; i++) {
      if (i in this && tester.call(that, this[i], i, this)) return true;
    } return false;
  };
}

Usage:

contains([{a: 1}, {a: 2}], {a: 1}); // true

Why to use JSON.stringify?

Array.indexOf and Array.includes (as well as most of the answers here) only compare by reference and not by value.

[{a: 1}, {a: 2}].includes({a: 1});
// false, because {a: 1} is a new object

Bonus

Non-optimized ES6 one-liner:

[{a: 1}, {a: 2}].some(item => JSON.stringify(item) === JSON.stringify({a: 1));
// true

Note: Comparing objects by value will work better if the keys are in the same order, so to be safe you might sort the keys first with a package like this one: https://www.npmjs.com/package/sort-keys

Updated the contains function with a perf optimization. Thanks itinance for pointing it out.