JPG or JPEG (Joint Photographic Experts Group)
Everyone's heard of a JPG, right?! JPGs generally have a smaller file size, but it comes at a cost. These files are compressed -- the smaller the file size, the more pixelation. These are not transparent; they will always have a white background. You should use a JPG when you're designing something for the web or when you want to have a small size that's great for emailing or storing.
PNG (Portable Network Graphics)
PNGs are also low resolution, but they can have transparent backgrounds. For that reason they're often used on the web or in places a small image is needed without the white background. This is not what you want for anything printed.
TIFF (Tagged Image File)
These are known for high quality and are used a lot in photography and graphic design.
GIF (Graphics Interchange Format)
This is a commonly used web image format, often for animated graphics like banner ads, email images and short videos. They are transparent.
This is a format available from high-end digital cameras. The data is high quality, uncompressed and unprocessed which provides a clean, high-quality image that one can edit in applications like Photoshop or Lightroom. The files can then be converted to JPGs or TIFs.
PSD (Photoshop Document)
These are files originally created in Photoshop that can contain layers and image adjustments. They're usually used to create and edit raster images, but it can also contain vector layers as well.
PDF (Portable Document Format)
PDFs are extremely versatile. They can display both raster and vector graphics, and is basically a universal standard that printers use for print production.
EPS (Encapsulated PostScript)
These are primarily used as a vector format, but they can include raster data. They are not used for photographs or artwork, but they are commonly used for vector images like logos.
AI (Adobe Illustrator)
These are files originally created in Adobe Illustrator and are primarily vector-based. They basically work like EPS files.