Most digital multimeters look like this:
1 = Display, 2 = Function switch, 3 = Transistor socket (optional), 4...6 = Test lead jacks
If you want to measure DC current, set the function switch to the DC current range you want to use. For example, if you expect to measure 1mA, set the switch to 2mA DC. If you have no idea what to expect, set the function switch to the highest DC range available and work down.
Having done that, we can connect the test leads. Mulimeters usually come with two test leads: a black one and a red one. To measure current, you need to connect the black test lead to the COM jack and the red lead to the A jack. Connect the other ends of the test leads in series with the load under measurement. If the current flows from red to black, you will read a positive value. Otherwise, a minus sign appears in the display.
If you want to measure AC current, set the function switch to the proper AC current range. Connect the test leads in series with device-under-test. Swapping test leads makes no difference (of course!).
Note: many meters have a separate jack for measuring high current. Usually the A jack measures up to 200mA. The separate jack will be labeled '20A'. This jack only works when the function switch has been set to 20A. Warning: the 20A jack is usually unfused! Overload may seriously damage your multimeter.
Tip: if you want to measure the current flow in a component, you'll have to connect the meter in series with that component. This means you may need to unsolder one end of that component. If the same current also flows in a resistor, you can simply measure the voltage across that resistor and calculate the current.
After current measurement, disconnect the leads from the meter. If you forget this and want to measure voltages again, you may cause disasterous shorts!