Dig under the side of the foundation that is sloping or sinking. Dig underneath the foundation until the footings -- the lowest parts of the foundation -- are showing. Dig as deep as you can underneath, exposing as much of the underside of the foundation as possible. The deeper, the better. Expose the underside of the foundation throughout the entire area that is sinking. The more foundation that is exposed, the more supports that can be placed under it.
Add some gravel to the bottom of your hole if you are still unsure about the soil. The point of digging deeper than the original foundation is to get to deeper layers of soil that are more stable. Often, unstable soil is the reason for the slope. If you live in an area with high bedrock, that is even better, since you will use this as leverage. Adding an inch or so of gravel will give the house some extra stability if you are uncertain about even the deep layer soils.
Install the foundation bracket. The main way of supporting a foundation is the use of brackets and hydraulic piers to dig even further into the soil. The bracket works a little like a car jack. Get the underside of your foundation. The hydraulic motor will then send its base deeper into the ground until it reaches either bedrock or well-compressed soil. The motor will then, ideally, serve to lift the house slightly. At the very least, it will not sink anymore. There is no guarantee that the house will lift. If the lower layers of soil are rocky or firm, this should work.
Use a helical system if the motorized pier does not work. The helix system is identical to the hydraulic pier, except the stem that gets driven into the ground has circular pans around it at intervals, looking a little like radar receivers. The point here is to add extra support for the house. If the entire side of the house is sloping, then you will have to dig out the entire side and use as many of these as necessary to lift the house.