<p>I've been reading about "web services" here on SO, on Wikipedia, Google, etc., and I don't quite understand what they are.  What is the plain English definition/description?</p>

<p>If I make a simple website using PHP that just, say, prints a random integer to the page... is this a "web service"?  If not, why not?</p>

Yes that is a simple web service.
Web services are really nothing more than a request/ response mechanism that allows a client to remotely access/ modify data. There are formal standards for web services (SOAP, SOA etc), but your simple page is a service too.
The main downside to printing it to a page is that your service would return HTML. Preferable data formats are JSON and XML, because most client frameworks (and server frameworks) are designed around using JSON and XML.
So if you modified your service to return:
```
```
<RANDOM>some random number</RANDOM>
```
```
rather than:
```
```
<HEAD>...</HEAD>  
<BODY>some random number</BODY>
```
```
then it would be more useful to most clients