What exactly does mAh rating means?
The mAh rating indicates the capacity of the battery in current (mA) and time (h). This means that a 3000 mAh battery should be able to supply 3000 mA during one hour. But there is more. It might be that the battery is only able to supply 1500 mA /h. This is indicated in the C value mentioned on the battery or at least in the datasheet. There is also a C value for chargeing the battery.
All this information can be found in the datasheet.
How does mAh rating tell the current supplied to the load? How does it vary with the load specification.
The mAh rating tells nothing about the current supplied to the load. The design of the load determines the maximum current that will be drawn from a battery.
Again our example. If the C=1 and the load takes less then 3000 mA then you can use the battery without any problem. If the actual load from the application is 2000 mA then you could use the battery in theory 1.5 h. If the load takes more then 3000 mA then your battery is overloaded and will get damaged.
Also, suppose a mobile phone's battery has 3000mAh, if i supply it with 1000mA, then does it mean that the battery will be fully charged in 3 hours?
If that was true it would be nice but there are always losses involved. So in practice you need more time. How much more depend on the (remaining) quality of the battery.
And will a 3000 mAh battery deliver a current of 3000mA for 1 hour to a load?
If that was true it also would be nice. But at the end of a discharge the voltage of a battery drops and might become to low. Modern LIPO batteries also don't like to be discharged completely. So the discharge time available is less then the basis calculation.
If you treat your batteries well you will stay more close to the theoretical capacity. Mistreating reduces the capacity heavely. Temperature also plays an important role. That is why cars have starting problems in the winter.