I want to run my internet router which has power of i12 V - 1A . Can I run it directly from a battery of 12 V -100 amp current. Will my device be damaged due to the high amperes of current. Or will it work fine despite this difference

 

Your device will only draw the current that it needs. Not drawing the maximum current that the battery can supply only means the battery will provide approximately +12V for a longer period of time before it needs to be recharged.

 

You mean I can run my device without any problem. there will be no damage to my device.

 

Correct.

 

Unless ur device shorts out and burns the cables.
A fuse is a good idea placed as close as possible to the battery

 

the problem might arise

1. if router has internal regulator and requires input voltage higher of certain level
2. if your router requires an input voltage strictly inside certain ±percent range (is what the SMPS supply)
   
3. if you mistakenly connect it with wrong polarity (usually the core is Plus , shell is Minus - for tubular connector - and it may be printed on router near DC input - i have seen and reverse polarity plugs/devices - not so often)

 

Your car radio, which takes a few amps, runs from your car battery, which can deliver several hundred amps, and it does that without a problem.
Why would you think your router is different from the radio?