Here is how to check the settings on a Vista laptop, and alter them.
To the left of the clock, right click the Battery/Power icon.
-Click on Power Options.
Find the plan you are currently using (It should have a filled in button) and press "Change Plan Settings"
Click "Change Advanced Settings"
Scroll down to battery, and take a look at those settings.
The one you are mentioning should be "Critical Battery Action".
Change "On Battery" to the one of your choice, either Hibernate, Sleep or Shut down.
(Note, Hibernate is recommended, but it sound like your computer is not completly hibernating as it would appear to be shut down).
Also make sure "Low Battey Notification" is set to ON, and "Low Battery Action" is set to the same as "Critical Battery Action" (so if you selected to Hibernate, put it to Hibernate as well. This gives your computer a buffer to do what it needs correctly incase it hangs.)
To awake from sleep mode, plugging in your computer should wake it up. This varies by computer manufacture however due to BIOS diffrences.
To awake it in general, moving the mouse, pressing a key on the keyboard, or pressing (BUT NOT HOLDING) the power button will turn it back on.
To awake from Hibernate mode, press the power button after the lights have turned off. It should appear exactly like it does when it is shut down.