If I use a VOIP provider, I don’t need any special telephony or Digium cards connected to my server, right? Just the NIC to connect to the Internet, right?
I’m thinking that my Asterisk would just talk to the VOIP provider through my Internet connection (I have 100Mbps both ways). I would virtually be using their cards instead, right?
that’s right, no additional hardware required. now run away before i kill you for your 'net connection 
But you will need an IP Phone, or an ATA converter.
Why would I need an ATA converter? What does it do and where to connect it to?
An ATA converter has a network connection on one end and a normal RJ11 phone line jack on the other. It allows you to use a regular phone for VOIP.
There is actually a third option and that is to us a softphone like X-lite and a headset on your computer.
But I will not be using the system to take or make calls myself. It will be used to facilitate calls like a middle-man. So I will not need an ATA or softphone, right?
right, just the people using your system would. The “end users”.
The short answer to your question is yes. I just did it. Took me a while as just like you see here. There is lot’s of information on how to setup with extra hardware. But if you just wanna find out how to install and run * or tackle a more ambitious projects like you describe, there are ways to do it without spending a cent, if you consider that you’re high speed connection and existing computer are regular expenses you have to deal with anyway.
I got frustrated because most of the docs would break out configuration into sections for Zap, SIP, IAX2. If you’re not careful to pay attention to the Zap stuff, even though you don’t have a zaptel card, you will miss something needed for the SIP/IAX2stuff.
I did breakdown and by a cheap IP phone. It’s been worth the $78 I spent but again, I could have used that $78 to pay for a tank of gas.
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I using PSTN/VOIP SIP gateway from Micronet SP5012 without any probs 