Difference between shunt resistant measurement circuits using op amps
As null has pointed out in the comments, 


  The first circuit is a differential amplifier and the second is a non-inverting amplifier.


You will be able to find many articles explaining their operation, characteristics and limitations in the standard textbooks and on the web.

Ground reference and floating

What may be worth pointing out is that the second circuit will only work when the sensing resistor is on the ground side of the load and connected directly to ground. The first one will work when the shunt is, within certain limits, floating with respect to ground. The location of the shunt in the circuit will determine which configuration will suit your application.

Voltage rail limits

Every op-amp has limits on the input voltage. These are given in the datasheet and determine the max/min allowable input voltages where correct operation is guaranteed. There are absolute maximum ratings which when exceeded will destroy the device. Pay attention to these.

The op-amp output will also have limits on the maximum and minimum output voltage it can supply. Again see the datasheet but you may need to learn what they mean too!

Isolation and safety

Both configurations as drawn are only suitable for low-voltage DC measurements as they are not isolated from the op-amp and micro and can not handle negative voltages.