For normal current measurements you put a small resistance (current shunt) in the circuit and amplify the voltage drop across the resistor, which can be done with an op amp, to give a voltage indication of the current. Another, more complex way which does not use a shunt, is to use a hall effect device and amplifier circuitry to detect the current.
Edit: For currents in the mA or below region you can direct the current into an op amp summing junction (minus input) with a feedback resistor from the output to the junction, to give a voltage output proportional to the current into the junction. The voltage polarity will be negative for a current flow into the input.