Hans,
A Browser is a piece of software that lets you use the internet by displaying web pages.
Different computer operating systems come with different browsers.
So the browser on Microsoft is Internet Explorer. The browser on Android is called Chrome, and the browser on Apple is called Safari. If you do not like the browser bundled with your computer, you can download another one such as Firefox or Opera. Some browsers work on all systems, while others don't.
A search engine is a web page that is on the world wide web. It catelogues all web pages, and lets you search words or phrases. Google.com is the biggest search engine, but there are others. So when people they they goodled it, they mean they searched for something on Hoogle's website in their browser of choice.
Google the company (not the search engine) have branched out from doing just search, to making their own operating systems, Chrome for desktop and Android for mobile. Those use the Chrome browser. The Chrome browser also has versions that run on Microsoft and Apple operating systems.