Anyone who regularly uses the public wifi at an airport, hotel, or cafe knows that being connected to a network and having internet access are not the same thing. Unfortunately, your Mac isn't great about telling you whether you're actually online or not, which can lead to some frustrating troubleshooting.
A free application Online Check solves this problem. It is a menu bar icon that turns red when you're offline. You can also ask it to send you notifications when your online status status changes. It's a simple program, but life saving if you're somewhere with an unstable connection. You can install Online Check from the Mac App Store or the developer's website.
How Online Check works
I used to work in IT, which meant spending a lot of time troubleshooting network connection issues. Usually the quickest way to test whether the internet connection is working on a Mac is to open the Terminal and type "ping apple.com," which will launch an attempt to connect to Apple.com; this will let you know whether your internet connection actually works. Online Check is just a menu bar app that does this for you on a regular basis.
Online Check's menu bar icon gives you quick access to the Mac's wifi settings, which is helpful while troubleshooting, and you can click a button to see a log of when your computer connected to and disconnected from the internet.
There's not a lot on offer in the way of settings. You can decide whether or not the application launches at login, and opt to only show the menu bar icon when your computer is offline (handy if you're trying to free up space on your menu bar). Basically, you can install the application and not think about it until you need it, which is honestly how most tools should work.