How can I benchmark JavaScript code?

Just simple way.

console.time('test');
console.timeEnd('test');

jsperf.com is the go-to site for testing JS performance. Start there. If you need a framework for running your own tests from the command line or scripts use Benchmark.js, the library upon which jsperf.com is built.

Note: Anyone testing Javascript code should educate themselves on the pitfalls of "microbenchmarks" (small tests that target a specific feature or operation, rather than more complex tests based on real-world code patterns). Such tests can be useful but are prone to inaccuracy due to how modern JS runtimes operate. Vyacheslav Egorov's presentation on performance and benchmarking is worth watching to get a feel for the nature of the problem(s).

Edit: Removed references to my JSLitmus work as it's just no longer relevant or useful.

Just adding a quick timer to the mix, which someone may find useful:

var timer = function(name) {
    var start = new Date();
    return {
        stop: function() {
            var end  = new Date();
            var time = end.getTime() - start.getTime();
            console.log('Timer:', name, 'finished in', time, 'ms');
        }
    }
};

Ideally it would be placed in a class, and not used as a global like I did for example purposes above. Using it would be pretty simple:

var t = timer('Some label');
// code to benchmark
t.stop(); // prints the time elapsed to the js console

Just time several iterations of each function. One iteration probably won't be enough, but (depending on how complex your functions are) somewhere closer to 100 or even 1,000 iterations should do the job.

Firebug also has a profiler if you want to see which parts of your function are slowing it down.

Edit: To future readers, the below answer recommending JSPerf should be the correct answer. I would delete mine, but I can't because it has been selected by the OP. There is much more to benchmarking than just running many iterations, and JSPerf takes care of that for you.