What file type should I save my GIS maps in?

In order to save a map in ArcGIS, you need to export it. You can export in different formats such as .pdf, .jpg, or .tif. The choice of the format depends on the final purpose for your map and the types of programs it will be used with (e.g. word processor, powerpoint, photoshop).  Each format has advantages and shortcomings. For more information, visit Digital image file types from Dr. Matthews of Wake Forest University for a  description of image files format and their differences.

Common image file format

*Table from ESRI ArcGIS Resources Help section

File format Description
EMF (Windows Enhanced Metafile) EMF files are native Windows graphics files that can contain a mixture of vector and raster data. They are useful for embedding in Windows documents because the vector portions of the EMF can be resized without loss of quality. However, since EMF does not support font embedding and is exclusively a Windows format, it is not commonly used as an interchange format between users.
EPS (Encapsulated PostScript) EPS files use the PostScript page description language to describe vector and raster objects. PostScript is the publishing industry standard for high-end graphics files, cartography, and printing. EPS files can be edited in many drawing applications or placed as a graphic in most page layout applications. EPS files exported from ArcMap support embedding of fonts so that users who do not have Esri fonts installed can still view the proper symbology. EPS exports from ArcMap can define colors in CMYK or RGB values.
AI (Adobe Illustrator) AI files are an excellent format for postprocessing in Adobe Illustrator as well as an interchange format for publishing. The ArcMap AI format preserves most layers from the ArcMap table of contents. However, the Adobe Illustrator file format that ArcMap writes does not support font embedding, so users that do not have the Esri fonts installed may not be able to view AI files with the proper symbology. AI exports from ArcMap can define colors in CMYK or RGB values.
PDF (Portable Document Format) PDF files are designed to be consistently viewable and printable across different platforms. They are commonly used for distributing documents on the web, and the format is now an official ISO standard for document interchange. ArcMap PDFs are editable in many graphics applications and retain map georeference information, annotation, labeling, and feature attribute data. PDF exports from ArcMap support embedding of fonts and thus can display symbology correctly even if the user does not have Esri fonts installed. PDF exports from ArcMap can define colors in CMYK or RGB values. For detailed information on exporting to PDF, see Exporting to PDF.
SVG (Scalable Vector Graphics) SVG is an XML-based file format that has been specifically designed for viewing on the web. SVG can contain both vector and raster information. Some web browsers may require a plug-in to view SVG files; older browsers may not be able to view SVG files at all. SVG supports font embedding, so users who do not have the Esri fonts installed can still view ArcMap SVG exports with proper symbology. ArcMap can also produce compressed SVG files. The file extension changes to *.svgz when this option is enabled.
BMP (Microsoft Windows bitmap) BMP files are simple, native Windows raster images. BMPs can store pixel data at several bit depths and can be compressed using the lossless RLE method. BMP images are typically much larger than formats such as JPEG or PNG.
JPEG (Joint Photographic Experts Group) JPEG files are compressed image files. They support 24-bit color and have been a popular choice for use on the web because JPEG file size is often substantially smaller than many other image formats. However, the JPEG compression algorithm is lossy and is not recommended for most map images, as line drawings and text or iconic graphics become blurred by compression artifacts. Thus, PNG is usually a superior format for map images. JPEGs exported from the data view in ArcMap can be generated with an accompanying world file for use as georeferenced raster data.
PNG (Portable Network Graphics) PNG is a versatile raster format that can be displayed in web browsers and inserted into other documents. It supports 24-bit color and uses a lossless compression. For maps, PNG is often the best raster format, since the lossless compression keeps text and line work legible by preventing the compression artifacts that can occur in JPEG format. PNG files also have the ability to define a transparent color; part of the image can be displayed as transparent in a web browser, allowing backgrounds, images, or colors to show through. PNGs exported from the data view in ArcMap can be generated with an accompanying world file for use as georeferenced raster data.
TIFF (Tagged Image File Format) TIFF files are the best choice for importing into image editing applications and are also a common GIS raster data format. However, they cannot be natively viewed by a web browser. ArcMap TIFFs exported from the data view also support georeferencing information in GeoTIFF tags or in a separate world file for use as raster data.
GIF (Graphic Interchange Format) GIF files are a legacy raster format for use on the web. GIFs cannot contain more than 256 colors (8 bits per pixel), which, along with optional lossless RLE or LZW compression, makes them smaller than other file formats. Like PNG, GIF files also have the ability to define a transparent color. GIFs exported from data view in ArcMap can be generated with an accompanying world file for use as georeferenced raster data.

For more  information on exporting map specifically for ArcGIS 10.1.