It TOTALLY depends on the type of site/service/product. I'm currently involved in two very different models:
ThatHigh.com - ad driven, completely free, entertainment. There is no chance to pay for anything (well, I sell stickers, but that isn't something that generates much money). This site pays my rent through advertisements.
Djangy.com - Heroku for Python. When we're ready for public beta, it will have a free version: basically enough to get something up and running. The paid features are scaling, addons, dedicated databases, etc. There's no way advertising would work here, for example. The numbers are a little tricky -- only a certain % of apps will be paid, so to break even or reach profitability means that the paying customers are supporting EVERYONE, which can be significant.
You can imagine many different models -- Facebook, for example, and twitter. Then there are subscriptions that give you "credits", and then there's something like Twilio that you pay for in advance, and then your account is refilled automatically once it reaches a certain threshold.
This trend of "pay for what you use" has some implications for the organization providing the value and charging -- namely, your product has to be ultra-efficient in order to compensate for the fact that your users are only paying for what they use, even though there may be lots of remnant resources that YOU are still paying for.
It's definitely getting interesting!