Craftylady's explanation is great about IP addresses

Heres an explanation that I used to give to customers when I was doing tech support on the phone.

Theres different kinds of IP addreses - Internal IP and External IP. I'm going to use a hotel as an analogy. Inside the hotel are a lot of different room numbers, and each is a specific location. These numbers are used by the hotel itself to know what rooms there are -- this is like an internal IP address. Each computer inside your network is assigned a 'room number' that makes sense within your network. Each of these numbers (IP addresses) needs to be unique, since you can't have more than one 'room' with the same number.

Now, the hotel has a street address as well, and this address is used by the rest of the world to know where that hotel is. This is like your External IP address, which is assigned by your Internet Provider.
Probabally a daft explanation, but its the general idea

Your router is generally responsible for assigning IP addresses to all the devices within your network (depending on your settings), and theres a few things that could be done to stop the conflict from happening, like assigning each computer a IP address and not letting the router give them one.

It might be worth bribing the nerdy kid on your block to come and have a look at your network for you. The conflict won't hurt anything, and doing the reset like you do will reset it, however its kind of a pain in the rear for you to have to reset everything all the time