Over the years there have been an assortment filetypes, all competing standards that decided their way was best, all generally have their pros and cons. A filetype is often developed with what it is to be used for in mind, older file types were generally more specialised but more modern standards incorporate a range of features; such as colour spaces (RGB/CMYK), colour indexing and bit depth, compression (lossy/lossless), and metadata (program/camera/location info), or vector files that can contain images within them.
To keep things simple, you can always ask the person who you are intending to send this to as to their preference; for print, it would generally be best to use either TIFFs or PDFs as both of these support the CMYK colour space, and PDFs support vector graphics (text or line art); for web design it is best to use JPEGs or PNGs, these both support the RGB colour space and provide you with options to balance quality and file size.
Also to note:
1. If you have large files which will not email due to file size limits, ask who you are dealing with for advice, or use file transfer websites such as Mailbigfile, WeTransfer, or cloud storage services (Dropbox or Google Drive).
2. Once you save an image with lossy compression, You CANNOT recover the lost quality as any discarded pixel information is lost, so make sure that you save a “master” copy (a TIFF, PNG, or a PSD (Photoshop Document)).
Common File types…
JPEG (Joint Photographic Experts Group)
The most commonly used file type on the internet, this employs lossy compression and supports a greyscale or 24-bit RGB colour space. First becoming a standard in 1992, it has been used widely supported and used across the internet where websites need to maintain a small loading time, whilst allowing the designer to decide on the acceptable quality of the image.
See also JPEG2000 (.jpf or .jp2)
A successor to the JPEG standard, it gives you the option of either a lossy or lossless compression, whilst also supporting all current colour spaces (supporting CMYK means that it is suitable for printing). Unfortunately, due to the complexity of the encoding and not really differentiating itself from JPEGs or PNGs, it has not seen much use outside of the digital TV/film industry.
GIF (Graphics Interchange Format)
Most well known for its ability to be animated (looping multiple frames) it has seen large amounts of use across the internet, and with being developed in 1989 (with multiple improvements since) and being one of the first image standards, is almost universally supported across all devices. But with only being able to support 256 colours, it’s rarely used on modern websites (with designers slowly adopting WEBMs, or video files if animation is required) and should not be used in print; as a vector graphic, or PNG, can now do what a GIF would be considered suitable for but at a much higher quality.
PNG (Portable Network Graphic)
A successor to the GIF, an open format developed in 1996 due to licencing disputes and a need to support more than 256 colours, has been widely adopted for saving high-resolution graphics or where the quality of images need to be maintained. Using lossless compression, it excels with images that have large areas of single colour and also as a “master” file when editing an image. It supports RGB 8-bit (256) colours, or RGB 24-bit (16 million) colours, as well the option for transparencies, whilst also containing gamma information for accurate colour reproduction.
Animated derivatives have also been developed, but rarely supported.
TIFF (Tagged Image File Format)
The most common and preferred image format used in print, with the current version (6.0) developed in 1992 but going back to 1986, its flexibility has gone from only supporting binary 2-colours (black or white), to greyscale, to 24-bit and 48-bit colours, RGB and CMYK. It also allows lossy (JPEG) or lossless (LZW or ZIP) compression, even with the option for no compression at all (although this is for compatibility, not for image quality) which can result in very large file sizes. This is commonly supported amongst image editing programs, illustration programs or layout programs, but has not seen much adoption on web browsers.
A format developed by Microsoft for the Windows operating system does not utilise compression (which leads to large file sizes) and can store information in various colour depths as well as support an alpha transparency layer. Whilst rarely used outside of very specific scenarios, due to been well documented with no licencing and free of patents will ensure that this will continue to be supported for years and decades to come.
PSD (Adobe Photoshop Document)
This is a format developed by Adobe to allow features present in photoshop to be saved and re-edited, it supports all colour depths and spaces that photoshop does, whilst also allowing backwards compatibility with older versions of the program. It is also supported by other Adobe software (a PSD can be placed onto an illustrator document) and some competing image editors. This is perfect for a “master” file as it utilises lossless compression, thus quality is maintained across editing and saving.
This is only a short list of the most commonly used file formats, there are many more, saving a file in photoshop can generally show how many there are, but if you stick to the common file types or just ask for assistance, you should not run into many issues.