Raster Images: Tagged Image File Format (TIFF)
TIFF is a raster "wrapper." Strictly speaking, it isn't a format, but a container that can hold several different formats of data. It is protected by a copyright currently owned by Adobe Systems, and it has never been fully standardized. As a result, some programs handle it differently than others, and there are occasions when a tiff written on one computer or by one application cannot be read by another. I have personally run into problems moving tiffs from PC to Mac installations of Photoshop, and also from my compositing software (Nuke) to Adobe Premiere on the same computer.
Advantages of TIFF include its ability to hold layer information, much like a Photoshop document, and multiple alpha channels. Many applications will misinterpret the additional channels, though, and there is really no way to know which channel will be selected as transparency, so it is advisable to include only a single alpha in a tiff intended for display. Some extensions to the Tiff format include support for high bit-depth images, up to 32 bits per channel (96 or even 128 bits per pixel), CMYK color, or even YCbCr (YUV) color.
Available compression algorithms that can typically be applied to a tiff include LZW, DEFLATE, and RLE. None of these do particularly well with photographic data; no better than PNG, really. As a result, there is not really any reason to choose tiff as a final export format over PNG. Tiff's flexibility is useful, but ultimately limits it as a format for interchange between artists and systems.
Raster Images: Truevision Advanced Raster Graphics Adapter (TGA)
Around the same time that TIFF was being developed, Truevision was developing another similar format for digital video applications. Truevision was eventually purchased by Avid Technology, which continues to support the format in its video editing software. Unlike TIFF, TGA is unencumbered by copyright or patents, making it a somewhat safer format.
Targa is technically inferior to tiff because it lacks support for layering and can use only one type of compression (RLE). Its advantages, though, are inherent in that simplicity; every application and system that supports TGA will render it the same way, making it very popular still in video production pipelines. It supports a standardized alpha channel that most applications will interpret properly as a transparency channel. In addition to digital video, Targa is also popular in video gaming, where it is frequently used for texture maps.
TGA's maximum color depth is 32 bits per pixel: three 8-bit channels for RGB and another 8-bit alpha. Use TGA when you need lossless compression, transparency, and wide compatibility.
Raster Images: Photoshop Document (PSD and PSB)
The Photoshop Document format is not an export format, so I have put off detailing it. I've learned a little bit about it recently, though, so I thought I'd throw it in here.
PSD, as its name suggests, is the internal file format used by Adobe Photoshop. Because Photoshop is so widely used, PSD files can often be read by other software, although few other programs will access all of its features. This compatibility is helped along by PSD's structure, which is a modified form of TIFF. The largest difference between the two is the lack of adjustment layers in TIFF.
The related PSB (PhotoShop Big) format is intended for Photoshop documents that exceed 2 GB in file size and/or 30,000 pixels in resolution (horizontal or vertical). I believe that it also has provisions for images with 32-bit float channels, to permit High Dynamic Range (HDR) imagery. I am relatively new to HDR topics, though, so I don't know all of the details there. Anyone who does is welcome to chime in. PSB does have a limit of 300,000 x 300,000 pixels, but no file size limit that I am aware of (except that imposed by the file system, of course).