That's a good question, and one that you'll likely get a very mixed bag of responses for.
Personally if its a small / medium business I go for the annoying persistent "why" questionnaire approach to draw out what they really want out of their website and/or SEO campaign, I keep on at them until I get to the real reason why they want it.
Many clients have a reason why they want something, but more often than not they don't know their self what their real reason is, so their initial thoughts actually could be a total waste of resources and creates a false persona of what they REALLY want. Obviously Id present it a little more structured than this, but rough ideas :
Qualifying the customer
1. What are your goals, KPI's and what would you like to achieve ?
a) Traffic increase
b) Sale/conversion/lead increase
c) local visibility
b) organic position increase for specific terms.
e) other ?
1b. Why ?
2. What are your expectations that your budget would retain and what is your ideal monthly budget without stretching yourselves ? ( we want to make sure they are in it for the long hall right ? )
2b. Why ?
3. What time scales do you expect to start seeing movements towards your outlined goals.
3b. Why ?
4. Who or what do you consider to be your main competition ?
4b. Why ?
5. Are you investing in any other marketing verticals ?
5b. What are they and are they working
6. Have you had SEO carried out before, could I see any documentation you've previously received and why are you no longer working with them ( cheeky, but can provide some valuable insight )
7. What is your knowledge surrounding SEO and Online marketing?
7b. How hands on and involved do you want to be ?
8. Do you have multiple locations ?
9. Do you have any developers maintaining your website, what kind of access can we have ?
10. What do you consider to be your USP ( unique selling point )
11. Do you have an idea of your customer demographic and targetting ?
12. Do you have analytics, webmaster tools, Google maps already setup.
13. In your mind, what do you see as the "problem" ?
I tend to do much of the research of their site and how their business operates before meeting them, so asking what kind of website it is to me is fluff, In a meeting im searching for confirmations of what I already know more than anything.
I ask "Why" with most questions to pull out answers that they were not really expecting to answer or didn't really want to get into, they might get frustrated initially, but when they realise you've just pulled teeth to get the answer you really want they tend to like the fact you've done something most others wouldn't with the attention to detail.
Eventually you get to the real reasons for wanting the service, Initially you may get answers like "Well, we sell blue widgets, so we'd like to rank for blue widgets, it makes sense right ?" but the answers you really want to get down are "Well my competition XXXXXX is spanking me everywhere i look and I believe ranking x,y,z is helping their business massively".
You can then go away armed with that info and formulate a plan which highlights their exact problem also outlining you can do even more to improve here, here, here and here.
No worries, when i have more time ill structure the post a little better and add some more info but wanted to post quickly while I had 5 mins.
By the time I agree to a meeting I normally have already qualified their budget ( so you know if its worth your time going to meet them ) and have qualified most of the info.
Meetings for me have always been a matter of getting to know the client, getting them to like and want to work with you and to finalise whats already been talked about. Most of the other info would have been discussed via email or phone prior to agreeing a meeting. ( I hate going into meetings blind )