New owner - can't turn on pool

If it was working when you turned the timer post in the past and now is not it probably is a breaker or GFCI at the main power center. One of the 4 grey flex conduits should be the input power line you need to find where that comes from, it should be in a separate power center with in sight of the equipment pad if it is built to code. If you don't see another box then it is likely at your homes main breaker panel.
 
Agree with atttech-2. Guessing the main breaker is tripped. You certainly need to know where it is and check it. Probably in the main breaker box with your power meter. The second grey box you found is for landscape low voltage lighting. I could hear it was powering the transformer on and off just listening to the video. The smaller black plastic box hanging by cords is probably a temporary outdoor light switch like used for Christmas lights that sometimes has a remote control to turn it on and off.
 
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