A very popular way is to use a very low value power resistor in series with the load you want to measure. As current flows through it it will induce an e.m.f (voltage) across the resistor. This resistor voltage can then be measured by an ADC in a microcontroller. The value will be linearly proportional to the current running through it. Using Ohm's law you can deduce that the voltage across the the resistor is equal to the current x resistance. As the resistor may be slightly off, the device might need calibration.
An issue with this though, you want the minimum voltage drop possible across the resistor. This reduces power loss, and minimises the effect you will have on the load. For this reason a very small resistor needs to be used. There are plenty of resistors out there for this purpose, known as shunt or sense resistors. This tiny voltage could be as small as 0.1v, or maybe even lower, way too small for a standard ADC to pick up reliably. For this reason There is a need for an amplifier, to multiply this voltage by 20 or 50 times. This enables the swing to be measured across the range of the ADC. So if you have a 5v ADC, and the maximum voltage across the resistor will be 0.1v, the amplifier will need to have a gain of 50. There are two main categories of current sensors like this, High side and Low side.
The above image shows the basic configuration of these two types of measurement. The difference is based off on the location of the sense resistor. Low side sensing is between load and ground, with high side sensing between power and the load. It shouldn't make much difference, the voltage across the Sense resistor will always be the same.
One reason for not using the the low side method is for the fact it is based off the ground reference. If anything between the power and the high side of the load is shorted, the current sensor wont pick it up. It is just one thing thing that you can't then implement into your design.
Also be careful when choosing the amplifier and the resistors you intend to use. There are many amplifiers on the market designed for this specific purpose. The TSC101 is an amplifier I recently included in a project, for this exact purpose. A high side current sensor, with a precision trimmed preset gain of either 20, 50, or 100. Adding in a laser precision trimmed power resistor to this, and there is an output for a microcontroller, a very simple current sensing application. for less than £2 in your application.
