Cloud computing, there are two major players - Amazon (Amazon Web Services (AWS)) and Microsoft (Azure). They have massive server farms running all over the globe, running clever software that means these farms look and act as though they are thousands/millions of individual PCs/servers to the outside world. You rent one of these virtual PCs (called virtual machines (VMs)) and only pay for the minutes/hours that you're using it - the cost starts when you press 'On' and stop when you press 'Shutdown'. There are no fees when it's turned off. You access the VM through standard desktop software similar to Teamviewer (called Remote Desktop).
You usually get to choose where your VM is hosted. For PIO purposes it will make bugger all difference if it's in Europe, US or Timbuktu as all the processing is done inside the VM and then a comparatively small amount of data (charts, EV figures) will need to be transmitted back to you, whereas if you were a corporate running your enterprise business software on it then you'd want it hosted as close to your users as possible to avoid screen lag.
Microsoft and Amazon can offer different types of VM for different purposes. Entry-level lower powered computing for developing code, testing etc. High CPU power for intensive batch programs (like PIO). High storage capacity for big databases etc. The more power you want, the more the per-hour rental costs.
The theory is that the horsepower behind the VM is elastic, so if you want to add more RAM or more storage or more CPUs then it's a flick of a switch to do that rather than having to physically install more hardware or even reinstall on a new machine.
So, for example:
https://azure.microsoft.com/en-gb/pricing/details/virtual-machines/windows/An entry level VM with 2 CPUs and 4GB RAM is £0.025/hour i.e. if you left it on 24/7/365 (but why would you?) it would cost £219/year.
A "compute optimised" VM with 32 CPUs and 64GB RAM is £1.02/hour.
A "compute optimised" VM with 64 CPUs and 128GB RAM is £4.11/hour.
So you open Remote Desktop, login to your VM and off you go. Install PIO, do whatever it does (run trees I believe), email the results to yourself and turn off the VM. Later that day/week/month, you log back in, turn the VM on and carry on. Once a month, Microsoft/Amazon send you an invoice.
A quick read of the PIO FAQ suggests it's a personal licence so one option to potentially reduce your cloud costs would be to install PIO on both your VM and local PC. Do all the preparatory work on your local PC that might take some time (create ranges, stack sizes, action etc.*) and then upload the resulting script to the VM and let it do the heavy computing work.
* I'm making assumptions here on how it works from a two minute scan of the PIO website.