As raster-based images go, gif and tiff are indexed color, so they are smaller/more appropriate for images with only a few colors, like flat-colored logos and such. Of the two, I had thought that gifs were smaller (based on some experimentation that one of my graduate professors did). I could not figure out why anyone would use a tiff if this were the case, but there must be something that differentiates them.
GIF images only support 256 colors (indexed color, as you mention), so for anything that need to use gradient effects, anti-aliasing, or anything other than a pretty small number of solid colors, GIF is unsuitable. GIF is a compressed format, so it is suitable for small, simple graphics online, but not for print publication.
TIFF, on the other hand, actually isn't a single file format, but a collection of them (well, it's a single format that can be extended in lots of ways). It supports indexed color, RGB, CMYK, and other color formats, it supports both compressed and uncompressed images, it supports a variety of different metadata tags for the image, images with layers, and so on. Basically every professional graphics program can read TIFFs, but some might supports certain features, and others might support different ones. That means that frequently, you get the "lowest common denominator" TIFF file, for greatest compatibility with different programs. The lowest common denominator is usually 32-bit uncompressed, which is why TIFF files come out so much larger than the equivalent GIF. You might be able to try to turn on compression when exporting the TIFF file, but that might make it incompatible with the publisher's systems.
The other option for web images, (though not generally used for professional publication) is PNG, which is losslessly compressed but supports 32-bit colors. This is good for the cases where you would use a GIF, but need smoother colors or multiple levels of transparency. Again, it's pretty much only used on the web, an older versions of Internet Explorer have some trouble viewing certain kinds of PNGs, but for most purposes it's better than a GIF.
For photographic-type pictures, with tons and tons of gradations of color, a jpeg creates a smaller file, and I have often seen people request them, since I think many tools are able to deal with them. I personally notice some odd color distortion around edges, which I think is due to compression inherent to the file-type... maybe? Did I mention I'm not much of a geek? But I think at print resolution it's hardly noticeable.
JPEG uses lossy compression, as opposed to the lossless compression used in GIF or PNG files, which means that it loses a little information. This loss of information is what makes the edges look a little funny. It means, however, that it is much better at compressing large, photographic images, and so is frequently what is used for such images. I've seen some print publications prefer JPEG, and some prefer the uncompressed TIFF. I'd recommend having photographs available in both JPEG and TIFF.
I did have someone ask for actual photoshop (or jpeg) files once, and I was at a loss as to why they asked for those, and did not list any of the indexed-color types - it was a shame, since I already had lots of things as tiffs, that Mar had generated. I mean, if they have Photoshop, they should be able to change the file type to whatever they want, right?
Yeah, it doesn't make much sense to ask for the original files; if you give them a 32 bit uncompressed TIFF, they should be able to do anything with that that they want.
for vector-based graphics, we go with eps files, rather than, say Illustrator files... That seems clear enough to me. If you can take them, then vector-based graphics are great, since they're almost infinitely sizeable without loss of resolution. Eps is the only generic vector-based file type I know (I'm sure there are more, but I think it's very widely used and accepted by many applications, yes?)
The other widely used and supported one is PDF. If you make both EPSes and PDFs, you should cover all of your bases, but just sticking with EPS will probably be fine.
I _think_ these will be used most often for magazine-sized pictures, with more than one image on a page - not so much for full page product shots, or posters, so I'm thinking about 3-4 inch on a side, depending on aspect-ratio. That seems plenty big enough to me. What do you all think?
That's probably a good estimate, but I'd say you should aim a bit high to be on the safe side. Usually it's better for them to wait a little longer to download but have a graphic that works, than have to contact you for a custom graphic.
Also, I'm wondering if I can use the "save for web" feature in Photoshop, which compresses the file-size somewhat (I think - I don't really know much about compression, here). I like it because it lets you see how big the file will be, saved out as different types. But will that compression adversely affect the image quality (significantly)?
I don't know much about the save for web feature in Photoshop in particular, since I don't use Photoshop. Save for Web features usually do one of a few things; they might make several different resolutions of the image, they might strip off Mac-specific information that will be lost over the web anyhow, they might compress the image, and they might interlace it (make it so that instead of the rows of pixels coming in order, you get every fourth row or so first, then progressively more and more detail, so you can see a low-res version of the image while it's still downloading). What you should pay attention to is what sort of compression it's using; lossless or lossy. If it's using lossless compression, you should be all set, since that won't lose any information, which means it won't create any artifacts. If it's lossy, you need to decide if it's worth the reduced quality. As I said, I would recommend having both an uncompressed (or losslessly compressed) version and a compressed version of your photos available, so whoever is downloading can choose which they want.