I'd give your client an RGB JPEG of your logo saved at 300dpi. Before anyone shouts, I know that screen based images are 72dpi, but this way the website designer has some flexibility with scaling it without losing detail and can then re-save the image down to 72dpi once the size has been agreed.
Unless of course, you know exactly the size that it is needed on the site in which case you can skip the middle step and save it at 100% at 72dpi.
You may be able to keep the size of the logo very small by saving it as a GIF file, but this depends on the complexity of your logo.
If your client requires the logo for print purposes then you should really be creating CMYK, Pantone and black & white/greyscale versions for him to use. I generally supply these as EPS for placing in Quark etc. and create JPEG versions for use within Word etc.
I don't go crazy as I don't want to give them extra pointless files they don't need, and also a huge variety will only confuse them for a start, but I think it's good to offer a selection that aims to cover all potential uses, so I provide:
Vector PDF Vector EPS 300dpi JPEG Three diff sizes of 72dpi jpeg
Then I advise them that they have free ongoing support and to come back to me if they need extra files or replacements.
Periodically I check client's online profiles and websites and if I catch sight of a blurry logo I send them an email telling them off for not contacting me and with the right file attached. I seriously do
I also need to save in 300dpi, but I'm not sure how on CS4. Save for Web I presume saves it at 72dpi and there's no JPEG option in Save As.
If someone could add a little note next to each of the above (plus the 300dpi) that I could send to the client to explain why they need each and who might use them then that would be fantastic, cheers.
The customer won't need the business card at 72dpi, you should supply them with the vector source file if you want them to be able to edit your design at a later date themselves, or, by using another designer, and a high res PDF version that is print ready.
The logo is the only item that is really needed in different formats, for website display, for print etc etc.
The files you send them of the card artwork should be without black edges (unless they are part of the design). The files should show trim marks and should have 3mm bleed (though by the look of it, there is nothing that bleeds off the edge.)
The reason you should only really be sending vector/eps/pdf versions of the file is that the printer can then remove the trim marks by deleting the relevent colour ink plate, leaving the grey and blue Pantone colours. Bitmap/JPEG files don't contain this specific colour info.
If your client decides to have them printed in four colour (CMYK) then the JPEG will suffice, but the printer can convert the Pantone ref into CMYK, but you will notice a shift in the colour unless you've been clever and issued the JPEG with a matched version of the blue pantone