Question: From my understanding your ISP will assign (sometimes dynamically) an IP address to your ***router.***

(Or is it assigned to the modem? [I'm starting to think it is.])

If it is assigned to the **router**, then this question follows:

What if I plug my PC directly into the modem via an Ethernet cable.

Does my computer have hardware that interprets the IP being assigned by the ISP the same way the router does?

If anyone can offer up some details on how this work that'd be great.

Thanks!

Answer: 
The automatic assignment of IP addresses happens via a standard known as the Dynamic Host Configuration Protocol. 

When a computer connects to the network, the DHCP client, generally a component of the operating system sends out a DHCP request and recieves an offer from a DHCP server. The offer generally contains:

1. An IP Address
2. A subnet mask for that IP address
3. A gateway that should be used for off-network requests (like those going via the internet)
4. One or more DNS server addresses so the computer knows where to send DNS requests.

### Connecting to a Router

When you plug your computer into a router or associate your wireless adapter with an access point, the computer will usually receive a DHCP response directly from the router, which may itself have requested an IP from the ISP for the interface the modem uses. it will then route traffic from the computer via it's interface connected to the modem where appropriate.

### Connecting to a Modem

When you connect your computer to a modem, the DHCP request generated by the computer may be handled directly by the modem itself if the modem is running a DHCP server.

### Static Address Assignment

If the modem is not running a DHCP server, you may have to staticly configure the client device's IP address, subnet mask, default gateway and DNS servers. If this is the case, the correct values will be giiven to subscribers, or entered by a service technician when the service is setup.

Side note: This answer deals with addresses for end-user devices. Your modem also has an interface to the ISPs network, the address for this interface may also be configured by a DHCP server on the provider's network, however this depends how the provider chooses to configure their network.
Your PC has software ([DHCP](http://en.wikipedia.org/wiki/Dynamic_Host_Configuration_Protocol) client) that can request IP settings to be assigned to it by the ISP the same way the router does.