Manual you set the time by some combination of up or down to the value you want. Push another key, and the time will likely start to count down, and the enlarger socket will energize.
Auto - you program in a sensitivity value to have the timer indicate the time that corresponds to the one you made the test print at, while postioning the probe in an area you want to match in the future. Usually you set it to the densest are of the neg that has tonality i.e. the lightest grey in the print.
Then the next print - stick the probe in the same sort of area and the timer will automatically set a time, regrdaless if you change the aperture, head height, etc. when you place the probe on the area where you want the calibrated tone to be reproduced.
Usually you make the measurement with no filters in the light path.
Auto 1 and 2 can be used to store values from session to session. Can be used when working with papers of different sensitivity.
I don't have the manual to know that these ideas are completely correct for you piece of kit, but the way I have described is the general way a couple of differnent integrated analyser/timers I have work.