Ah! That answers a question I was going to ask when I'd got to the end of this topic - Do laptops have a place where I could plug it in? I have a Dolphin 'PEN' - which has a copy of my Supernova 12 software on it, so if my computer is out of action again (as it was for three weeks last autumn because of a local telephone exchange broken link), I can take it to someone else's computer and be able to use it. I tried it on my son's computer at Christmas, and he had to download something first, but then it was really good.
You bring up several different issues. I'm going to try and separate them out, so be patient with me if I seem to go over things you already know.
USB (aka Universal Serial Bus) is a connection standard developed for the purpose of attaching different kinds of devices to computers. From your point of view it's a place to plug stuff in. These devices include (but are not limited to) various kinds of data storage media.
Any computer that is much less than a decade old, laptop or desktop, should have one or more USB ports. The standard has changed since it was first implemented, but in general the older devices should work just fine in the newer computers, if perhaps somewhat slower.
A USB flash drive (or "thingie"

) is a solid state storage device which plugs into a USB port in a computer and looks like any other data storage drive to the computer's operating system.
Many programs are taking advantage of the portability of flash drives to allow people to carry personalized versions of the software around with them, so, for example, your web browser will still have all of your preference settings even if you change machines.
Sometimes, in addition to carrying data the flash drive can also serve as a copy protection device which insures that only the legitimate owners of a proprietary (You gotta pay to use it.) program are allowed to use it.
ETA: (I hit the "Submit" button by mistake.

)
Your Dolphin software is doing both of the above. The Pen is a portable version of the accessability software Dolphin sells (kinda pricey, too) installed on a flash drive thingie. Your son downloaded an adjunct program provided by Dolphin called "Interceptor" which Dolphin offers for free to allow a host machine to use the software on the flash drive.
Here is a link to the download page for Interceptor. You'll have to make sure which version is right for your Dolphin Pen.
You could download a copy of the Interceptor software and carry it around as well, which would let you use your Dolphin Pen on machines which aren't connected to the internet. You might be able to store it on the Pen itself, but I didn't look hard enough to determine that. You can certainly store it on a separate flash drive ... thingie.
ETA I've been just wondering whether to buy a small laptop.
Sure. If you go with a used one the prices are insanely low. Much less than the license fee for your Dolphin software.