GIF - Good for websites, also good for transparency if a design has areas that you don't want printed, not the best as far as detail goes though. Raster image type.
TIF - Don't use it much, believe it's good for a high quality photo, but believe that a high quality jpeg or png is used in its place now in most applications. Believe it supports transparency as well. Raster image type.
JPG - Jpegs, the most commonly used image format in my opinion. Good for websites, but I believe it is a lossy format, so some quality is lost in its compression scheme, also it doesn't support transparency. Raster image type.
PNG - Lossless compression good for high quality images and transparency, although the files can still get to be quite big in my opinion, more and more browsers are supporting it, but it hasn't become a standard yet. Raster image type.
AI - Adobe Illustrator, great for vectors, can also have bitmap and vector combined. Generally doesn't work too well with anything other than Illustrator, unless you save it to an older version. Vector format that allows Raster images as well.
EPS - Encapsulated Post Script, great for vector as well, lot better supportwise than AI format by other applications, although some details/formatting may be lost when converting from AI. Vector format that allows for Raster images as well.
In case you aren't aware, raster images are your typical everyday pictures that you see on webpages, your camera outputs, etc. You can do all kinds of neat things to a raster image with programs like Photoshop, however once a raster image has been made, it cannot be resized to a bigger size without some quality being lost along the way. Vector images however can be designed and it doesn't matter if they're 1"1" or 100'x100', they retain their quality because they use a combination of shapes to create the image instead of pixels. Vectors may be used in combination with raster images when doing wide format print and cut with vehicle decals or the like. So, I'd say keep all the items, but you'd probably use the Jpeg for your website, inhouse stationary, etc, and the EPS file for any kind of posters, window decals, vehicle decals, t-shirts, etc.