Selections are critical to many, many things, and there are many ways to create them. Only one or two seem to be well known. Let's look at most of them, with a focus on what we use them for most — removing a background from an image. Also be aware that for most tools, Shift, Option/Alt, and Command can change the function of almost any selection tool. Usually Shift extends a selection, Option/Alt reduces one, Shift-Option/Alt intersects a selection, and Command/Ctrl gives you the Move tool, to move a selection around.

Marquee: the box or oval for quick and dirty selections. Easy, useful often.

Freehand, Polygon & Magnetic Lasso: for freehand selections, polygons, and freehand selections that snap to edges. I don't like any of them very much. Use them to create a rough outline if you have to.

Magic Wand: Often used, often misunderstood. It selects areas of similar colour, but not in the safest way possible; it will often select half of an edge in a sharp, inappropriate way. Don't use it too much when separating an object from its background — its edges will be ugly.

Color Range: very, very good for selecting large areas of similar colour. Many complex objects, a pain to trace around in any other way, will respond well to Color Range selection. You can use it to select one or more colours (Shift-click) and nearby colours (use the Fuzziness slider) with good-looking edges. Works less well with noisy, ugly sources and when the object shares many colours with the background. Even if it shares a few, start from a good Color Range selection, and move into Quick Mask mode for further touchups.

Quick Mask: Great. Make a rough selection with Lasso or Magic Wand, then switch into Quick Mask mode by pressing M. Now use the brush — fairly hard — to paint in the correct edges. To remove the red overlay rather than add it, press X to exchange foreground and background colours, and brush again. Use smaller brushes for smaller areas. Don't use fuzzy brushes — you want a clean selection when separating backgrounds. Don't select shadows on the background, we'll add new ones with layer styles afterwards.

When you're done, press M to return to normal operation, and flip the selection around if you've selected the background, not the object (Command/Ctrl-Shift-I). Jump the object to a new layer and hide the original layer. Erase as needed. If you're missing part of the object, Undo back to the Quick Mask state and fix it up.

Make sure you've got clean edges by creating a temporary background layer of bright, contrasting colours (red and green Clouds works well). You'll find it hard to spot problems with a white or a transparent grid background. When you go to the final stage, the embedded drop shadow layer style will show any problems even more clearly.

PS. A graphics tablet will help greatly with Quick Mask, letting you fine-tune brush width with pressure. Use it to adjust size of the brush rather than opacity, though.

Layer Masks: same principle as Quick Mask, except you don't actually destroy the original image data at all. Use Layer > Add Mask > Reveal All for starters (other options should explain themselves). You'll see the new mask in the Layers palette, and can show it through the Channels palette. Paint on it by selecting it in Channels, or more easily, clicking the mask thumbnail to the right of the regular image thumbnail on that layer, in the Layers palette. Easiest of all is just to press \ (backslash) when editing the layer. Your mask pops up just like a Quick Mask, only entirely non-destructively. Very nice indeed.

Extract: great for removing an object from a background, and sometimes easier than masking. Duplicate the layer, then select Edit > Extract. Use the highlighter to trace the edges of the image roughly. Use Smart Highlighting (turn the checkbox on) if the edge is well defined — it will find edges more accurately. When done, use the bucket to fill in the selected area. Press Preview (not OK). Now use the Cleanup Tool (C) to erase extra bits around the edges of the image, holding Alt/Option to fill sections in — it's like the Quick Mask tool but with safety snapping. When it's all spiffy, hit OK. Should have nice clean edges, too.

Paths: used in print, and sometimes for cleaning up rough edges. Also very good as a starting point when cutting out a tricky object from its background. Transparency in Photoshop has shades; most layer style drop shadows have only partial transparency, for example. And paths don't, just inside and outside, on and off. However, paths are vector based, and that makes them powerful. They can be used in print to mask out unwanted areas of the image, in Quark Xpress, for example.

But by using the pen tool to draw an accurate path around an object (an art in itself and worth practising) then creating a selection from that path (feathered by 1 or 2) you can create an accurate, smooth, cutout quite efficiently. You'll need to use the shortcuts detailed below to help you. You can also try creating a path from a selection, if you just want to smooth out an existing selection. Also use Option/Alt-click on the New Path button to access the tightness of the path — you'll usually want 0.5 for screen resolution smoothing on complex objects.

Selection Modifiers: Invert, Expand, Contract, Border, Feather, Smooth. If you need to fill in a few holes in your selection, and can accept a little smoothing around the edges, you can Expand, then Contract the selection, or just choose Smooth. To select a border, choose Border — though it tends to fuzz out a little. Feather is good in some situations, when you need to blend something seamlessly with the background. Don't use it all the time; it's not usually what you want when compositing. Inverting your selection (Command/Ctrl-Shift-I) is needed when you've selected the background rather than the object or vice-versa. In Quick Mask mode, because you're drawing the selection, just use the normal Invert command (Command/Ctrl-I) to do the same thing. Sometimes it's easier to paint the object red, sometimes clear. Either works well when you can invert, and putting the object onto a New Layer via Copy is less destructive than deleting the background from a layer.

Layer Transparency: Command/Ctrl-click on a layers or on a path to select the transparency of a layer or path. Very handy when creating fills or when creating transparency from grayscale images. Be careful, though: don't try to duplicate a layer by selecting its transparency and then Jumping it to a new layer. The edges won't entirely come through. If there are any soft edges — and there will be — you've just selected 50% of a 50% visible pixel. It won't come across entirely. Similar things come into play when creating alpha images outside of the PSD format — like, say, for Final Cut Pro.