Capra Demon is a boss you'll encounter as you explore Lower Undead Burg in Dark Souls.
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Capra Demon boss strategy
As soon as you traverse through the fog gate you will want to weave right and then left- up the stairs. Alongside the boss are two dogs, who you need to get rid of as soon as possible. From here, kill the two dogs to face Capra alone.
One of the most difficult things in Dark Souls is fighting multiple enemies in a tight space. With the fast pace of the dogs and heft of Capra's greatswords, you will want as much wiggle room as you can possibly get before focusing on the actual boss.
Hopefully, with the dogs out of the way you will be able to face Capra alone. Note how the stage is built here, you do not want to get caught in a corner, or in the arches on the right. Being here will make you incredibly susceptible to Capra's greatsword attacks. It is useful at the start in order to get around the dogs and Capra, but use it for the at purpose alone. There are also more nooks, with the tight space meaning that you may not be able to dodge many attacks at any one time.
This means, that you will be reliant on your 100% Physical damage blocker Heater Shield (Seriously, go pick one up from the merchant in the Undead Burg if you have not already.) From there, you will be able to defend, keep an eye out on our endurance however, as you will often find that if you get stunlocked against Capra, it will very very easily kill you.
Onto it's attacks, Capra has a fair few, the dual sword slashes left and right, meaning that you don't want to hang around the sides of its torso. In addition to this, it also brings both of its swords together and does a slam downwards. This is unblockable, so be sure to not get stuck and get stunlocked by it's followup side-slash. If you try to keep your distance from Capra, you will find that it also has a jumping slam, so try to roll out of this one, too.
Capra is actually weak to fire, so while you can use Firebombs and Fireballs, we wouldn't really recommend it due to windup and casting times. If you do choose to do this, ensure you've made your way up the stairs for a bit of distance against the Capra. There is a better way of killing the Capra, but it is going to be pretty risky.
First, you will want to kite the Capra over to the alcove next to the stairs, then you want to sprint, go up the stairs and do a plunging attack. When you do this, you will be able to take off a decent chunk of its health. From there, repeat this process, unless you find that the Capra follows you up.
If this happens, take it on, ducking in and out with a fast weapon to ensure you don't get hit by any of its attacks. The real key to this fight will be how effectively you will be able to utilise the staircase. Ensure you manage to get there, and then attack with whichever method we used above.
This is a big sticking point for many first-time players of Dark Souls, so if you are finding it pretty difficult, upgrade your weapons and be sure to get some decent armor in order to tackle it better with a bigger advantage. Once you manage to take the Capra Demon down, you will get some souls, and a key to The Depths.