Networking – Convert home phone wiring to Ethernet

I am trying to wire two Ethernet jack ports into a trailer outback so two computers can plug into the wall at the same time. I have one long CAT6 cable with an rj45 connector on one end that is plugged into the router in the main house, and loose wires on the other end.

OK.

Do I need to wire two jacks into the other side of this one cable and plug it into two ports in the router?

Short answer: yes.

Longer answer:

CAT6 (and cat 5 and ..) cables have 8 wires. 4 of those for 10/100Mbit Ethernet. All eight are used for gigabit speeds.

This means that you can split one 8 wire cable into two sets on both ends and run two 100mbit connections over it. You can do this either by creative wiring both ends, or you can use existing plugs similar to these:

This will effectively wire your network like this:

This is likely not officially supported, but it works well enough.

Note that you cannot run gigabit speeds over these cables anymore. You no longer have the 8 wires per connection needed for that.
If you have any electrical power in the trailer (and I presume you do, if for nothing than to run the computers off), then the proper answer becomes a 'do not do this, but a switch instead'. 4 port switches are dirt cheap, do not limit your speed down to 100Mbit and will save you a lot of time.

If you do no have power and are running one or more laptops on battery, then consider these cheap pre-made plugs over building your own cable. Keeping things standard is a good thing.

First of all: Yes you can, but it won't make you happy.

10Mbit/s Ethernet uses 2 pairs and works quite well over short distances with the kind of cable shown, you just need to crimp an RJ45 plug on it (using the pairs 1/2 and 3/6).

Now 10Mbit/s is not very fast, so you might want to go alternative routes. Assuming The idea behind using this cable is avoiding a recabling, you might want to consider Ethernet over powerlines, this works amazingly well in many home settings, the adapters (Devolo and TP-Link have worked for me) are cheap as chips.

EDIT

As requested in the comments, a few more details about the needed plugs:

Ethernet typically uses RJ45 plugs with 8 pins (1..8) carrying 4 pairs of wires. For historical reasons, these pairs are assigned to the pins this way: 1&2, 3&6, 4&5, 7&8. Older Ethernet connections use only the pairs 1&2 and 3&6.

So for a straight-through cable you need to connect pin 1 of one end with pin 1 on the other side, same with pins 2,3 and 6 and for a cross-over cable you need to connect 1-3, 2-6, 3-1, 6-2.

If the wires in the cable are paired (i.e. always two are twisted together inside the cable's protective mantle) honor this pairing when connecting the cables.

The mechanical process of putting the plug on the cable is called "crimping" and it is easily done with cheap "crimp pliers". If you don't have one, I recommend you just cut a cheap Ethernet patch cable in the middle and solder the two resulting "pigtail" half-cables to the ends of the old cable, using the orange/orange-white and the green/green-white pairs.