I've been working with web services for nearly 4 years now. Although people will argue this next point, I am a business person, not a technical person. My Business was construction engineering first, and although my business is of technology now, it is still fundamentally important for me to understand how to translate the ecosystems like those of human relationships into the world of technology relationships...enter web services. This is the underlying technology that is going to let me securely connect many disparate systems in many different places in a more efficient (less IT intensive) way. If you are looking for more info on integration, see my multi part series on the topic, otherwise, continue here...
A great example of web services in action is the way Facebook works with other social services like Wordpress or Twitter. I can make a post on Wordpress for example and it shows up on Facebook. I could make a Tweet on Twitter and have it show up on Wordpress. Three completely different services in three different technical domains from three different companies all working together. Excellent! I get all my info wherever I am, don't really care where it came from or how it got there.
An AEC Tech Space example would be entering information in Autodesk Revit, then having it travel through Horizontal Glue (now Autodesk 360 I believe), and end up in Proliance. This of course takes some implementation and the more in line software vendors are with NIST cloud standards the better this will work. The reason for that is the NIST cloud definition talks about self service and programmability. If both these are in place, then you do not need software vendors (although they may be the choice after all) to approve how you achieve this type of connection and you are free to compare different ways or do it yourself if you have the skills.
Some other examples of web services in action are using an Office Business Application (OBA) with a line of business (LOB) application like Prolog, Contract Manager or Viewpoint as an alternate interface. A user can work in the Excel based OBA from anywhere and be directly connected to Prolog (wherever it may reside) via the Prolog Connect web service. This gives users the power of excel while keeping in compliance with Prolog business rules. This also eliminates any import/export issues since the OBA is bound to the Prolog Connect web service. Further, you will not have any data corruption problems from some custom excel application that might connect direct to the Prolog database and circumvent the business rules. (Note that although Meridian does not brand it as Prolog Connect, the actual web service is called a Prolog Connect and is installed as such.)
There are several technical flavors of web services, but at the end of the day as a business person you just need to know the example above and then apply it to your requirements of mashing up applications and data into useful dashboards, entry forms, and other interesting and useful methods of accessing information. The other key thing to look for when purchasing software is that it has open web services that do not lock you or your team into engaging any specific company or proprietary solution to help you make it work.
The reason web services matter vs. just any old application programming interface (API) is it gives our community a secure way to connect applications over the Internet based on some standards. Customers can rely on standard easier to use integration points, Software Vendors have less support issues than older integration methods, and Service Providers can openly use the application connections to dream up up broad variations on ways for people to interact with technology.
If you want to get buried in the technical basis of all this, visit Wikipedia.