Welcome to the PF. There are a number of ways to measure the value of an inductor. The simplest, of course, is to us some instrument like an LCR meter. Those instruments impress an AC waveform (typically 1kHz or 10kHz or 100kHz) across the inductor, and measure the resulting current.
If you don't have an LCR meter or other AC impedance-measuring device, you can do it youself with a signal generator, a resistor and an oscilloscope. Use the fact that you will get a 45 degree phase shift through the resistor-inductor series combination when the reactance of the inductor equals the value of the resistor. Are you familiar with how the reactance of the inductor varies with frequency?
An inductor has a reactive impedance, which can interact with other impedances in a circuit and provide an AC voltage drop. A real inductor also has an associated DC resistance (DCR), which will act like any other resistor and provide a DC and AC voltage drop when a current flows through it. You usually will chose an inductor that has a small DCR compared to its reactive impedance at the frequencies of interest, though. Make sense?