What's a hot spot?

A hotspot is a physical location where people may obtain Internet access, typically using Wi-Fi technology, via a wireless local area network (WLAN) using a router connected to an internet service provider.

Put another way it's portable because it taps into 3G and/or 4G cellular networks, just like a smartphone does, and then wirelessly shares its data connection with other nearby (within 30 feet or so) Wi-Fi-enabled gadgets—typically, between three and 8 depending on the device and the carrier.

What good are they and why would I want one? 

You can create 'hot spots' with a device you purchase (see below) or you can create one using your iPhone and certain equipped iPads.

I have an iPhone.  Can I use that as a "hotspot"?  The Short answer is yes.  Go to >settings>Personal Hotspot>

How about using an iPad to create a hotspot?  Again the short answer is yes, if you have a data plan on the iPad.  If you don't know what this is you probably don't have one.  But, if you create a hotspot on your phone, you can connect with your iPad!

on your phone
If using your computer you see this:
What is the difference between using my phone to create a hotspot or purchasing a device from Verizon, AT&T or someone else?  The short answer is that using a phone hotspot to connect your computer or iPad to the internet is a "short time" solution.  That method uses your data plan.  If you have 'unlimited' data - great, no problem. 
However, choosing a below device might allow you to drop your WiFi provider all together and perhaps save some money.  It is also "portable" - you take it with you.  Wherever you go it goes.  You will always have a secure WiFi.  In the car, at you friends house, using your personal wiFi instead of one at the hotel you stay at.   
These devices also have an internal battery and will stay charged for some period of time like during a power outage or in a hurricane evacuation.  They will continue to work as long as cell phones work.