I came across batteries with _mAh_ rating, which says mA amount of current can be supplied for an hour; if we pull more power more discharge for less time. Some batteries are rated with voltage and output current; does that mean its current output is  fixed?  
For some batteries its output current rating is not mentioned but mAh rating and voltage is specified; does that mean current can be discharged according to the resistance in circuit?

I have a GSM module that needs 9v and 1 amp current to start transmission; it works with 12v 1.0 amp dc adapter but when I use a 9v dc battery with no indicated current rating it starts but turns off as soon as I try to make a call. Can I use a mAh rating 9v battery so that the GSM module takes current as it needs? If not, what can I use to take 1 amp current from the DC battery? Is there a way? I'm not a electronics engineer, I'm a software guy who doesn't know about electronics. I'm working with bus tracking project for my college, but unfortunately I couldn't find help for electronics side.