Layers
The layers palette looks like this:

The layering system in Photoshop is very simple - one on top of the other, and the bottom layers are by default obscured by those on top.
Buttons
Look at the buttons on bottom. From left to right, we have the Layer Styles button (Which is a whole different post... Don't worry about that one yet.), the Layer Mask button, the Set button, the Fill/Adjustment layer button, New Layer, and Delete Layer.
The last two are pretty self explanatory.
Fill/Adjustment layers change the attributes of the layers below them. For example, if you put an "Invert" layer (hold on the button for the menu of options) above other layers on your image, the layers below your new F/A layer will appear inverted, whereas the ones above will retain their normal coloring. This can be a powerful tool; if you have many layers and want to apply the same effect to a portion of them, you can put a F/A layer above them. There are many options for this type of layer, and I'll get into those in more detail some other time.
The Set button creates a Folder in the layers palette, so that you can organize your layers into a heirarchy. I almost never use it, but it can be nice when working with a LOT of layers.
The layer mask button creates a mask attached to your layer, a seperate image that governs the transparency of the main image on the layer. If you want the layer to be transparent or translucent at a certain spot, you simply paint the desired area black (for complete transparency) or grey (for a translucent effect). The darkness of the color influences the transparency.
Layer Styles, as I said, is much too useful to relegate it to a part of this post. I'll cover those later. Layer styles apply special effects to a layer, and are very, very powerful and useful.
Blending Modes
Blending modes are another important aspect of the layering system. They are, basically, the way that the top layers interact with the layers below them. These are also affected by the opacity slider on the top right of the menu. On the image above, the Blend mode menu reads "Normal." This is the default, where colored areas are opaque and transparency means see-through. When you move the opacity slider, the image fades away to nothingness. Simple enough, right? Now, let's review the uses of the other modes.
Dissolve: When you move the opacity slider, the image goes away pixel by pixel, with the remaining pixels staying at full opacity.
Darken: Simple enough - the darkest parts of the two layers show through. Any white areas covering black areas on the lower layer will become, in effect, transparent.
Multiply: Similar to Darken, but less of a strong effect.
Color Burn: Similar to Darken, but preserves coloring better.
Linear Burn: Similar to Color Burn, but a stronger, sharper effect.
Lighten: You guessed it! The opposite of Darken; light colors show through while dark colors are shoved aside.
Screen: Similar to Lighten, but smoother and less strong.
Color Dodge: Opposite of Color Burn.
Linear Dodge: Opposite of Linear Burn.
Overlay: Puts the color of the top area onto the bottom with respect to the luminosity of the bottom layer - lighter areas recieve more of the color.
Soft Light: Like Overlay, but, well, softer...
Hard Light: Puts the color on the bottom layer with little regard for luminosity.
Vivid Light: Transfers the color in a very strong way, makes it bright and sharp.
Linear Light: Similar to Hard Light, a bit sharper.
Pin Light: Puts the color down on the darker areas of the bottom image, leaving the light ones alone.
Difference: Inverts portions of the top image that are above lighter parts of the bottom image.
Exclusion: Like difference, but leaves the middle-luminosity range of the bottom image alone.
Hue: Transfers color in the parts of the bottom image that have the greatest amounts of color.
Saturation: Transfers the texture, not the color, of the top image onto the most colorful parts of the bottom image.
Color: Blends the colors of the two layers.
Luminosity: Projects the light and dark from the top to the bottom, leaving the color out.
Whew... sweatdrop
A couple more things - you see those buttons that are labeled "Lock:"? Those are really useful. The checkerboard one on the left is the Transparency lock. It prevents you from drawing on the transparent sections of the image, useful for preserving outlines.
The brush, the pixel lock, prevents changes to the layer.
The position lock prevents layer movement.
The last button locks all three.
Now, how do you use this information? Make pictures! Play around with the blend modes, with masks, see what happens... Try overlaying patterns onto photos, photos onto patterns, try using the transparency lock and painting over text. If you are unclear about anything, just ask and I'll explain more.
That's all for now... Next time - Layer Styles!