The mount connects the engine to the airframe or fuselage. Other important secondary features are to distribute the weight of the engine, while at the same time spreading the torque and vibration generated by the engine. And all of doing that under g-forces from turbulence and other pilot induced load factors.
Most aircraft with exactly the same type / manufacturer of the engine can usually use the same mount, and for Lycoming and Continental models its either a conical or dynafocal mount. Most diesel and Rotax engines have their own specifically designed mounts and method of installation.
Designing your own mount is also an option, although you will be required to calculate and demonstrate the strength needed for all tubing under ultimate positive and negative G loading.
Most mounts are made from tubular steel chrome-molybdenum (4130) and they are welded together. This is a lightweight and very strong construction. After welding the structure is sandblasted and powder coated in a bright color (white would be perfect), which makes it easy to spot any cracks should they develop (personal experience).
Most four cylinder engines today use mounts that look pretty much the same. The engine is bolted to the mount at the back or at the underside of the crankcase (some Rotax 912/914 engines are installed in such a way).
This is by far the easiest mount to fabricate. It has four attach points for the engine and usually four points to bolt the mount to the firewall. The mount points are parallel with the firewall, so there is no awkward angle when installing the bolts and shock mounts to the engine. Its all rather straight forward and really easy to do.
A drawback is that vibrations and engine torque are transmitted through the frame so this is not ideal compared to a dynafocal mount.
These are the best and do a perfect job of cushioning the vibrations and movements from the engine. Lower cockpit noise is the result with less pilot/crew fatigue. One drawback is that they are more difficult (expensive) to build / construct and install. As you can see from the image the engine is held in four attach points (located in a ring) and these are under a certain angle and point to the center of gravity of the engine.
During welding this angle must be held in perfect alignment or else the four bolts will not fit when installing the engine in the mount. You will need a steel jig for this.
There are two types of dynafocal mounts: nr.1 and nr.2. Type 1 is used to Lycoming engines up to 180 hp and the type 2 is used in the IO-320 and IO-360 model engines from Lycoming.
These are quite different compared to the conical and dynafocal mounts. With this model engine is mounted using four points underneath the crankcase and then hang onto the firewall. You will see this sometimes with Rotax or certain diesel engines.
Some aircraft designs call for an engine mount that imparts a downward or upward thrust relative to the waterline of the aircraft where others offset the engine to the right or left. The aircraft designer decides how much during aerodynamic calculations and takes engine power and other factors into account. This will compensate for the left turning tendency the aircraft has with a right turning propeller.
The engine is not bolted directly onto the mount, this would result in vibrations from the engine to the aircraft. In stead, stiff rubber shock mounts of varying strengths and thicknesses are used. This dampens the vibration and movement, giving a much smoother flight and running engine. Any airframe fatigue due to vibration is also no factor with this type of construction.
Creating your own mount requires that either you are a very good welder or that you know a good one. To get all alignments perfectly you will need to construct a jig from rigid steel with holes in it so it simulates the firewall and engine attach points. The dynafocal is the most difficult type as the engine attach points need to be at the correct angle.
Some kit aircraft manufacturers offer an installation kit which includes a mount for the engines they support and sell. If you go with a totally different engine than advised with the aircraft kit then you are on your own, unless there is another builder or group of builders who already went this way. It pays to investigate this with your local experimental aircraft chapter.