A lot of officers are looking for information on how to resign, and unfortunately the Army doesn't make this easy to find (I can't imagine why…). Thus, I've posted my resignation letter (sanitized) for download in the downloads section of ArmyAdvice.org. I also want to make sure people have a reasonable expectation of the timetable involved, so here's a summary of how things went for me (admittedly some time ago):

6 January 1995 – Enlisted as E3 w/8 year commitment (ETS 5 Jan 2003)
25 July 1997 – Received 2LT Commission – commitment automatically extended to commission date + 8 years (26 July 2005)
20 July 2005 – Sent resignation letter, with effective date of 26 July 2005, return receipt requested.
Early August – Received confirmation letter was received; called HRC to confirm and verify all was in order. Spoke with a Major there who confirmed it was. He told me to expect it to take 4 to 6 months.

Once/month – called HRC, spoke to same Major (I got his direct extension and email address the first time), confirmed things were moving and they were not waiting on me for anything (and hadn't lost my paperwork)

20 December 2005 – Received Honorable Discharge orders and certificate in the mail. Eventually my records on HRC's website were updated with these as well.

So, all told, it took about 5 months from the date I submitted my resignation letter until I received my discharge papers.