Amps, amp hours, volts...I'm a little confused here

This electrical stuff is pretty confusing to me. 
Do amp hours = the length of time something can run on those amps? Is it like miles per gallon, i.e. at 14 amp hours you could use 1 amp for 14 hours, or 7 amps for 2 hours, etc?
I am trying to figure out how to make this information understandable to me in a way that is useful to me.

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