Very nice story btw, sort of like a fairy tale in the computer world

Let's first analyse your situation from what I understand it is... (arrows indicate traffic from the internet to your computer, of course traffic actually goes both ways, just doing it like this for simplcity)
Code:
DSL modem <------- DSL line from ISP [phone line]
|___> your computer
|___> your sister's laptop
(Some modems have a built in router, you should check up on this first of all.)
Let's get down to business now. You have various types of networking equipment for different purposes. I'll just do a short run down of the ones in question.
A router - a router is what the name suggests, it
routes network traffic according to the address of the network packet. In its most basic application (my house for instance), it's connected to just two networks. This is usually the case for home DSL clients.
Code:
DSL modem/router <------- DSL line from ISP [phone line]
|
|
V
home network
| |_ > computer A
|___ > computer B
In this case it routes traffic from one network to the other. My DSL modem has a built in router and firewall. In this case the router routes traffic from one network (the wide area network [WAN] of the ISP) to the other (the local area network [LAN]). In more complicated cases, the router is connected to several networks (ie. internet routers) and routes between several.
A hub/switch - a hub is a device designed to connect several computers in a network. A switch is the same thing, only more efficient. A hub will broadcast a packet from computer a to all nodes, the packet is rejected by all those with an address not matching, accepted only by the intended recipient machine. A switch will instead send the packet only to the right place.
Code:
home network
| |_ > computer A
|___ > computer B
In its most basic application, a hub connects two computers across a network. Each computer has an ip address on the network (not to be confused with internet ip addresses!!).
Note: you need a hub/switch to connect three or more computers. To connect only 2 nodes you can use a cross linked network cable instead but then if you want them both to access the internet, one would have to act as a router for the other.
I don't know how clear an explanation that was but I'll get right to the point now. You have two options.
1. Use one computer as router
Needs:
- a cross linked network cable
- another network ethernet card
What to do:
- get a 2nd network ethernet card for your computer and install it, connect your computer through that socket to your sister's using the cross linked cable
Code:
DSL modem <------- phone line
|___> your computer
|___> your sister's laptop
Pros:
- cost
- simplicity
Cons:
- only 2 computers can be connected this way
- the computer serving as router must be active at all times when the 2nd wants to access the internet
2. Use a hub/switch
Needs:
- a hub or switch (3 or 5 ports will do)
- possibly a router unless your DSL modem has a built in router
What to do:
(in case you bought a router)
- plug the cable from the modem into the router
(otherwise start here)
- plug the cable from the router into the hub/switch uplink port
- connect each computer with a network cable (category 5 utp is most common in Norway) to a vacant port in the hub/switch
Code:
DSL modem/router <------- DSL line from ISP [phone line]
|
|
V
router (if not built into modem)
|
|
V
hub/switch
| |_ > computer A
|___ > computer B
Assumption:
- the router has a built-in dhcp server (most routers do I think) to assign ip addresses, otherwise you'll have to set them up manually
Pros:
- more computers can be added to network at any time as long as there are ports free in the hub/switch
- network is always up provided hub/switch is on and router/modem is/are also on
Cons:
- complexity
- cost
Oh and going wireless would mean pretty much the same thing (though only possible with option 2), but would be using a wireless access point/router instead of a regular router.