A browser is a program that allows you to use the internet. Familiar browsers include Safari, Firefox, Internet Explorer and Chrome. Not much thought goes into choosing a specific browser most of the time for most people. They all bring you to the same internet, right? Well, if you use a wide variety of websites for your class or yourself, you know that some sites work better in one browser rather than another. The trick is to learn which browser to choose for a given site. Often, this is just trial and error. There's also a very good chance that your students will be able to tell you which browser will work best for what you want to do. Before asking students to do any work online, of course it's always wise to try out a couple of browsers in case one doesn't work well.
Something that many teachers often forget about is the fact that browsers offer different services. For example, Chrome provides simple youtube video downloads. Safari on the other hand works best with Apple products. For example, you can send pictures directly to iPhoto from most websites just by right clicking. Firefox has a reputation for being one of the most secure browsers out there. It also has a video downloading extension, but it's a bit more cumbersome than what you get with Chrome.
With all browsers there is the option to save bookmarks and settings in a place other than your computer. Chrome gives you the option to store your information online. Safari keeps everything in iCloud. You can store bookmarks for Firefox and I.E. as well with special add-ons.
Even if you've found a browser that you love, consider trying some of the others out there. You may be pleasantly surprised.