Can someone please hold my hand a little to get me to understand them?

A netmask is ANDed to the IP address before it leaves the computer and goes to a router, right?

I don't really understand networking although I use it. Learn a little most days. Here is something I found with a search that might be helpful:
http://www.pointz.com/spudpage/subnets.html

It takes a bit of wading...

That's right. The ANDing of the mask with the destination ip determines the destination network number. This is used by routing points to determine where to route packets. Routing devices will have a direct connection to a few networks - packets for any destination network that isn't found are sent to the default route. Some number of the most significant bits of the IP address represent the network number and the rest designate the host. The mask will contain binary 1 values in each bit position that represents a network number and 0s on bits representing the host.

First, you must understand some binary math. Given the binary '1011' with the mask '0010' what happens when you AND these two numbers together?
Your result is '0010' -- everywhere there is a one-bit in your number with a corresponding one-bit in your mask, your result will contain a one-bit as well.

When applied to an IP address, say '172.25.30.21' and the net mask of
'255.255.0.0' your mask has one's on the left end giving a result of
'172.25.0.0' [REMEMBER -- an IP address is four decimal numbers representing four, 8-bit values, separated by periods.]

What the netmask actually says is

Given any IP address, apply the netmask (using AND) and which ever bits remain tell you the "network" portion of the address. The other IP address bits -- the ones that went away -- tell you which host within that network.

I hope that this helps. Check out the O'Reilly book on network routing for more than most people ever want to know.

Cheers,
~~~ 8d;-Dan