Explorer intermittently runs rough, idles @ low speeds

2004 Ford Explorer Base, 110K
Need a little help with this conundrum! This car is driven 500 miles/month or less, and these "episodes" occur about once a month.

At no-to-low speeds, the engine will begin to vibrate and sputter. Then the Check Gauge and Battery lights will come on; the sputtering and vibration will get progressively worse; and eventually the engine and car idle/die completely. During one of these "episodes", the engine will not deliver much power while pressing the throttle. If it is turned on/off a few times, it seems to run OK for at least a while (long enough to get home, anyway) and won't act up for maybe a few weeks.