Newbie Question

I’m completely new to the idea of a PBX, so please bear with me if my
question seem very unenlightened.

I’m looking to hook a computer up as the gateway for all phone calls
coming into my office. I would like to write some custom software to
log all of my calls as they come into the office.

Can asterisk help me do that?

Asterisk can replace any PBX you have (or link with it), and comes with Call Detail Reporting out-of-the-box so you can record details of all calls that go via Asterisk.