Shorter version:

const monthNames = ["January", "February", "March", "April", "May", "June",
  "July", "August", "September", "October", "November", "December"
];

const d = new Date();
document.write("The current month is " + monthNames[d.getMonth()]);

Note (2019-03-08) – This answer by me which I originally wrote in 2009 is outdated. See David Storey’s answer for a better solution.