Let's take my site as an example here. Most of my site is HTML content I designed from scratch. i.e. I thought up what to say and how to present it. There are also a few "backend" programs (like the search engine and the page upload scripts). I've also got things that handle mail sorting forgoes along with generation and presentation to some degree, but is more done "after the fact" instead of on the first go-around.
Information Processing is where I group together anything that isn't strictly HTML or simple JavaScript. It will really depend on what level of interactivity your customer wants on their site. It could be as simple as reading and responding to email people send to a specific address (but perhaps the customer can do that themselves). It could also mean writing a whole order placement system where the information is submitted on a webpage and gets processed with a program so the different pieces of information get where they need to go automatically.
The level of information processing that can be done will vary depending on your skill level and what the ISP will and won't let you do. In my case, I can write or use just about any program I'd like since my ISP gives me full CGI access and shell access to the web server machine. Most ISPs don't do that unless you run the website on your own machine or use their pre-defined CGI programs.
Processing site logs will be something that some people will be interested in. You can see how many people visit when and from where. There are programs that can make this easier, but some people write their own scripts to handle the work.
System Maintenance is anything that doesn't strictly involve the website content or information processing for the site. This means things like dealing with the ISP on a myriad of technical and non-technical issues, backing up the content and programs necessary to run the website, making sure the system the website runs on stays operational (particularly if you own the server). If you have a domain, it means occasionally dealing with the InterNIC and a lot of other non-web related things like email, FTP servers, and so on.
While this last thing sounds daunting, in most cases, it usually isn't. But it can encompass a lot, particularly if you're hosting the site for the customer.
In terms of how to charge them, most web site designers either charge:
Make it very clear the kinds of services you can and can not provide.
Define the responsibilities very clearly. Perhaps this means writing up
a contract. Don't commit to anything you don't feel comfortable doing.