Shutter speed, 'stop', exposure, EV
The best place to start is probably the shutter speed as it's one of the most important aspects of photography, the shutter speed is simply how long the shutter is open for to expose the digital sensor (or film). The correct shutter speed is vital, a very high shutter speed is required when shooting a subject moving at speed, when shooting in low light a much slower shutter speed is needed to absorb the available light however without a tripod or similar this can lead to a blurred shot. If too slow a shutter speed is chosen for a selected scene then the picture will be overexposed with everything looking too bright as the shutter has let in too much light. The opposite is underexposure where the shutterspeed was too high and didn't let enough light in leading to a picture which is too dark. The problem is auto-exposure systems cannot tell exactly what you are taking a picture of which means they normally work out the exposure/shutter speed from the whole scene whereas it may be one person or a certain object you want to show correctly. This is where EV (exposure value) compensation comes in useful which is shown as a +/- symbol on most cameras, this allows you to use an automatic shutter speed but change the value slightly to suit your circumstances. For example if you are shooting in bright sunlight you may want to set negative EV compensation which will increase the shutter speed and prevent overexposure.
A frequent term in photography is a 'stop' which refers to either doubling or halfing the shutter speed - if you set your camera to underexpose by one stop it will double the shutter speed whereas overexposing by one stop will half the shutter speed. Similarly increase the ISO from 100 to 200 increases the shutter speed by one stop as does widening the aperture from f-stop 2.8 f-stop 2.0.
Aperture, f-stop
Aperture refers to the amount of light the lens allows into the camera which is used to calculate the shutter speed. It is measured in f-stops with a high f-value meaning a wide aperture letting a lot of light in and small f-values refer to very small holes letting a small amount of light in. F-stops are set to indicate exactly one stop of light between them, changing from F2 to F2.8 for example will half your shutter speed as the lens is letting in half the light. Changing from F2.8 to F.4 lets in half as much light again meaning the shutter speed is also halved again. Lenses are shown with their maximum aperture value which for many zooms is variable depending on lens (expensive top of the range lenses have a constant aperture), for example a 12-60mm F2.8-F4.0 lens means that when the lens is at its widest setting of 12mm the maximum aperture is F2.8 but at the zoomed in setting of 60mm the maximum aperture is F4.0, one stop slower. This is just the maximum aperture though and it is possible to choose a number of smaller apertures to suit the conditions, anyone who has watched Stargate SG-1 will be familiar with the iris which protects the Stargate against unwanted visitors - camera lenses have an almost identical iris inside them which closes to specific sizes for each aperture. When the lens is wide open at its maximum aperture the blades forming the iris will not extend at all allowing as much light as possible through the lens. At the minimum aperture the iris blades will have closed almost entirely only allowing a very small amount of light through the very small circle. To be clear shooting wide open refers to using the lens at its maximum aperture whereas 'stopping down' means changing the lens to a smaller aperture (great f-stop number).
Older lenses used to have the aperture value physically selected on the lens itself however modern lenses keep the aperture wide open regardless of the chosen f-stop while composing the shot to allow as much light as possible making it easier for the user to see the scene and help the AF mechanism. The moment the shutter button is pressed the apeture blades close down to the correct setting then once the picture is taken they immediately retract to leave the lens wide open. SLRs particularly will normally have a preview button which when pressed with close the aperture blades to the chosen aperture to give the user an idea of what the scene will look like.
Aperture values don't tend to affect compact cameras very much due to their small sensors (explained later), it's a more important feature for digital SLRs where aperture values have a much greater affect on the depth of field (also explained later) and the quality of the image. Many lenses are sharper with less vignetting (dark corners) when they are stopped down to around F5.6 or F8.
The f-stop of a lens is calculated by dividing the focal length by the aperture value (the width of the opening), one very important aspect of f-stops is that they are the same when referring to different lenses - this picture is a good example:

This shows the Panasonic LX2 against its L1 sibling, clearly the LX2 is tiny in comparison to the much larger SLR. Despite this, both cameras have the same effective focal length of 28-100mm and both have a maximum aperture of F2.8 at 28mm despite the fact the SLR lens is far bigger than LX2. When shooting the same scene both cameras will produce the same shutter speed at 28mm F2.8 although the depth of field will be different.
Focal length, equivalent focal length, zoom
Focal length refers to the angle of view the lens produces, very low focal lengths have a large field of view ('zoomed out') which will show a large part of a scene whereas high focal lengths will have a very small field of view ('zoomed in') showing a very small portion of a scene possibly a large distance away. In the film days it was simpler in that there were a limited amount of different types of film which all cameras shared, 35mm was a very popular size for both point and shoot cameras and larger SLRs. Digital cameras have no such restriction which means their sensors can be any size the manufacturer chooses which has led to a huge variety of different sizes even in very similar looking cameras. The problem this causes is that the actual focal length translates to different fields of view depending on the size of the sensor, 6mm on a compact camera produces a normal field of view, 7mm on an Olympus SLR is an ultrawide field of view on a 35mm SLR this is a very extreme field of view only available as a massively distorting fisheye lens. This is where the term 'equivalent focal length' comes from, the idea is that since many people are familiar with the field of view for a 35mm camera manufacturers translate the actual focal length of the lens to what focal length would produce the same field of view on a 35mm camera.
Back to the picture above with the Panasonic LX2 and L1, as mentioned both cameras have a similar equivalent focal length - the L1 lens has a 28-100mm range and the LX2 is slightly longer at 28-112mm. The actual focal length can be found by looking at the lens itself, the LX2 lens is 6.3-25.2mm and the L1 is 14-50mm. Conceptually it's easier to use the L1/14-50mm lens to explain where this equivalent focal length comes from, the 4/3 sensor in the L1 is almost exactly half the diagonal size of a 35mm sensor. If you imagine the L1 has a 10MP sensor and a 35mm camera also has a 10MP sensor it means that the 35mm camera (ignoring the slightly different aspect ratios) will produce a similar 10MP image with a 28mm lens. Alternatively if the 35mm camera has a 14mm lens mounted, cropping out a 4/3 sized section from the middle would produce the same field of view as the L1 with its 14mm lens however the resolution in this case would be lower.
Referring to equivalent focal length, these are the rough translations of the focal lengths:
14mm - Ultrawide angle
22mm - Superwide angle
28mm - Wide angle
35mm -50mm - Normal
100mm - short telephoto
200mm - Telephoto
500mm+ - Super telephoto
The zoom of a lens refers to the difference between the widest focal length and the narrowest which is commonly used for compact and bridge cameras but less so for SLR cameras. It's a slightly misleading term although the fact it produces a quantifiable number makes it a popular feature to market. The reason it is misleading is because you don't know where the lens is starting and ending, a 10x zoom could refer to a 28-280mm focal length which is wide to telephoto, 35-350mm which is normal to telephoto and no wide angle or even on an SLR it could be 100-1000mm which is telephoto to super telephoto. When considering a camera it's always worth checking what zoom range you are getting, personally I prefer to have a bit of wide angle at the sacrifice of some telephoto range as you can always crop however some people prefer to have as much range as possible.
Depth of field
Depth of field is the distance from the nearest point in focus to the furthest away point, which are respectively in front and behind of the focus point the camera has been set to. This means that a picture with large depth of field will have most of the scene in focus whereas a picture with shallow depth of field will have the subject in sharp focus but the rest of the scene blurred by being out of focus. The lens aperture is one of the main factors which governs the depth of field however the focal length (actual, not equivalent) and how close the focus point is to the camera also affect the depth of field. Choosing a wide aperture with a high focal length and focussing close to the subject will produce a very shallow depth of field whereas a small aperture (high f-stop), short focal length and focussed on a further away subject will mean that most of the picture is in focus. For example, if I put a 7mm lens on my 4/3 SLR then focus on an object 10m and set the lens to F8 the depth of field will be close to infinite. However if I mount a 50mm macro lens and focus it at 20cm even at F8 the depth of field is virtually non-existant, the field of focus is paper-thin.
As the actual focal length is used rather than the equivalent focal length compact and bridge cameras have large depth of field even when shooting wide open as their focal lengths are less than cameras with larger sensors. For general use this can be an advantage as it means even if focus misses slightly the picture will still look fine however for those who like the isolation effects by having a shallow depth of field this is considered a disadvantage and one of the reasons why people choose SLRs. Consider three cameras, again the LX2/L1 above although this time with a 25mm F1.4 lens and adding in the Nikon D3 (which has a 35mm sized sensor) with a 50mm F1.4 lens. You want to shoot a portrait shot of someone focussing the camera at 3m, 50mm focal length and with a shallow depth of field. Firstly with the LX2 the actual focal length is around 12mm and the widest aperture F3.5 - with these settings the depth of field will be 4m which is really too much for this type of shot. The L1's 25mm can open quite a bit wider to F1.4 allowing for a depth of field of just 60cm which is far more suitable for a portait shot. Finally the D3's 50mm lens similarly wide open at F1.4 has a depth of field which is smaller again at 30cm.
I've seen the question a few times how to get shallow depth of field with compact cameras for which the answer is unfortunately you can't really without post processing to blur out the background, using a compact camera at macro ranges will give a very shallow depth of field but when shooting macro you usually want as much depth of field as possible.
John