Google & You Tube did a survey some years ago, and found that less than 5% of people knew the difference. So you're definitely not alone if you’re not sure!
Here's an explanation, with a breakdown of the main differences. You'll see why each is vitally important in it's own way, and the different choices and options you have available...
Search Engines
A Search Engine is essentially a dynamic website that holds a huge index of all the web-pages that it thinks are worth listing. It's kind of like the index at the back of a reference book. The bigger the book, the longer it would take to find what you wanted if you just had to flip through.
It's estimated that Google, for example, indexes over 100 billion sites and 1 Trillion pages now - just imagine trying to find what you want in that lot! So Google uses some very fancy algorithms (and according to recent publications, the most powerful computer array on Earth,) to pull out a list of what you want - according to the keywords that you type in, and in order of importance or relevance.
Of course, this importance and relevance is somewhat subjective, which is why we often have to go through 3 or 4 pages to find what we really want. But you get the idea. Google is essentially 'just' a website that indexes an awful lot of other websites, although they're getting progressively more clever at working out what you actually want.
There is more interesting information on market-share and worldwide volume of searches here : https://www.smartinsights.com/search-engine-marketing/search-engine-statistics
Browsers
A Browser is quite simply a piece of computer software that interprets web-page code. i.e. it displays web-pages according to the languages they are created in (and there are lots of languages, with HTML & CSS being the base layout scripts). It's an advanced display tool, although much of the fancy decisions and data look-ups are done at the web server, before the page is sent out to you to be displayed. With modern JavaScript and AJAX etc., the amount of page interactivity (without refreshing the webpage) has increased exponentially though.
Without a browser you couldn't look at web-pages. The most popular browsers are Google Chrome, Mozilla Firefox, Microsoft Internet Explorer/Edge (the blue windows-based swirly 'e'), Opera & Apple Safari. There are actually over 200 different browsers available, and some are very obscure, but the top 5 are all we really need to worry about.
Mozilla Firefox and Microsoft Internet Explorer were the most well-known browsers on the PC for many years, but Google Chrome has now gained a huge amount of ground. And as a matter of fact, during early 2011, Chrome overtook Internet Explorer in terms of worldwide users. The vast majority of new users of Chrome have moved from the Internet Explorer camp, with the user-base of Firefox remaining reasonably constant; although they've lost some ground too.
As of July 2018, Google Chrome now completely dominates the world with around 80% of users, Mozilla Firefox has around 11%, MS Internet Explorer has dropped hugely to 3.5%, and Safari & Opera have around 3% and 1.5% respectively.
Google Chrome & Firefox’s popularity come from their ease of use, and the variety of useful plug-ins and add-ons that are available. They're also highly customisable, free to download, and available for both the PC and the Mac. From a web design & development stand-point, these are is the browsers of choice.
Mozilla Firefox can be downloaded from : https://www.mozilla.org/en-GB/firefox/new/
Google Chrome can be downloaded from : http://www.google.com/chrome
Most Windows operating systems have Internet Explorer/Edge available as standard (the blue swirly ‘e’ logo). As of version 9 (which arrived in early 2011,) MS Internet Explorer is now much improved.
Safari is the standard browser that comes preloaded with Mac OS, and is also seen on all iPhones & iPads. There’s not much to say here, except it’s loved by much of the Mac community; and in-keeping with all things Mac, it’s simple, elegant and just plain works! Many Mac desktop users are now migrating to Chrome or Firefox though, because of its benefits.
Opera doesn’t have a big following on desktop computers and doesn’t come installed as part of any operating system, but is widely used on mobile platforms. You’re likely to see this on many different mobile phones; like the Android platform (along with Chrome).
W3 Extra Information Link : http://www.w3schools.com/browsers