Yes, the alternator does spin all the time. It is ran off of your cars belt system. To further elaborate- open your hood. (Car OFF! ) Now locate the belt system on your car- if it is rear wheel drive- it will be right in front; if it is front wheel drive it will be to one side. To find the alternator- simple look for a pulley that has a fan looking device directly behind it.
You can also test your cars output voltage with a Multi-Meter. Easiest way is to connect the the meter directly to the battery- posative to posative, negative to negative; and set the meter to Volts DC - the meter should then display voltage at rest (not running) and voltage at idle.(running) At rest- you should have a reading of anywhere between 12.3 and 13.1 volts DC. At idle you should have a reading of 13.8-14.7 Volts DC. Anything deviating considerably from this should be cause for concern, as it would indicate a problem with either the battery, alternator, or voltage regulator.
take it easy,
-zane