Auto levels is the same as going into the individual RGB channels in levels, and sliding the highlight and shadow sliders to the ends of the histogram in each channel. It's making the darkest pixels 0, and the lightest pixels 255 in each channel.
Auto contrast deals with a luminosity histogram (all three RGB channels averaged together). It's the same as sliding the shadow and highlight sliders in a little past the outside edges of the luminosity histogram. It makes the almost darkest pixels 0, and the almost lightest pixels 255.
Levels allows you to adjust tones by setting 3 points (shadow, middle, and highlight), and then it averages out the histogram/tones between where you set the points. Curves allows you to set and adjust as many points as you want. With this you can have more control how PS is averaging the histogram between the shadows, and the midtones, and the highlights.
I set my curves display up so the shadows are in the lower left corner, and highlights in the upper right corner. I switch the grid to 10 x 10 squares (press alt and click on the grid, or something, to do this) so it's like the 10 zones of the zone system tonal scale.
When you start out the line runs straight diagonally from shadows to highlights; every point on the line corresponds to the same tone on the tonal scales running on the left side vertically, and the bottom side horizontally. If you take the cornermost point in the shadow corner, and slide it to the right horizontally, that's the same as sliding the shadow pointer to the right in levels. If you take the cornermost point in the highlight corner and slide it to the left horizontally, that's the same as sliding the highlight pointer to the left in levels. You can put a point, or points, anywhere on the line. Moving the point up makes that tone lighter. Moving the point down makes that tone darker. Leaving the point in the middle on the center diagonal is leaving the tone as it was when curves was opened. The curving line is a graphic representation of the computer averaging the tones between the set points.