The brightest is obviously the Sun. The second brightest is the Moon. The Moon is so bright that it casts clear shadows at night, you can read by it, and even through a modest telescope it can make your eyes water. So why do so many people think you can't see it during the day?

Perhaps they think it is washed out by the Sun. It isn't; if it's up and more than about first quarter it can be seen without any problem if you know where to look. Perhaps they think that it literally is only up during the night, and not during the day. This cannot be true; it orbits the Earth, making a complete circle once a month or so (the word 'month' actually comes from 'Moon'). It orbits in roughly the same path that the Sun follows on the sky, so sometimes it must be between the Earth and Sun, putting it up in the sky at the same time as the Sun, that is, during the day.

I think the answer is more subtle and more telling of how many people live their lives: they simply don't notice what goes on around them. The Moon is easy to see during the day, if people would simply look up! How many times have I seen beautiful rainbows and breathtaking halos around the Sun with multicolor sundogs and found that no one else has noticed? Most non-astronomers I talk to are shocked to find that they can see planets and man-made satellites with their unaided eyes on almost any clear night. But all they need to do is look up and see.

Phil Plait's bad astronomy does good work, clearing up misconceptions, and making good sense:

Now, bear with me: if summer started at the moment the Sun hit its peak, then every day of summer would mean the Sun gets a little lower in the sky. This to me doesn't make sense. You want the Sun to hit its peak at summer's midpoint, meaning summer would start 1.5 months earlier. That way the Sun would be climbing steadily higher until the peak of summer, then start to fall. This would in turn mean that the Sun hits its nadir (lowest point) in the middle of winter, and not at winter's beginning! See?