In general, it's not illegal for you to save pictures from a Google image search on your own computer for personal use. However, what you do with that image is subject to legal regulations. For example, putting the image up on your small business website could land you in trouble with the authorities over copyright infringement. That could lead to heavy fines and, in serious cases, even jail.
Images on publicly available websites can be saved for your own personal use. Personal use means that you're not using the image on your own site or making printed copies and distributing them to others. Google doesn't own the copyright to the pictures that appear in image search results; the individual site owner does. To save a Google image, you can just right-click your mouse on the image and select "save as."
Some images online are licensed under Creative Commons. This form of licensing usually permits certain forms of sharing and use of copyrighted pictures. In some cases, you can use a picture on both personal and commercial websites -- provided you clearly attribute the creator. Each license will show what the creator permits. However, the only way to check if the image is licensed by Creative Commons is to click an image in Google image search and visit the source website. Some photo sharing sites, such as Flickr, allow you to search Creative Commons photos.
Copyright and Fair Use
An image doesn't have to have a little copyright symbol or have legal information beneath it to be protected by copyright. So if you see pictures on Google image search that don't have a copyright symbol, that doesn't mean you can use them wherever you like. Saving them to your computer and using them privately is generally their only legal free use. In some rare cases, you can use copyrighted images under "fair use," but the laws for this are complex.
Even if an image turns up in a Google image search, it doesn't mean it's legal in your state or country. Some images, such as extreme pornography or images of child abuse, are illegal to download and keep on your computer. While such images rarely turn up in an image search, it is possible they will appear on an unfiltered search. Google provides a way to report possibly illegal images for removal.