Grafting is a process of making a piece of a plant grow by attaching it to another plant. This process is often used to attach a tree that is a good bearer of fruit to another that has strong roots, combining the best qualities of two plants. It can also be used to have different but related kinds of fruit growing on the same tree.
The graft in the picture has been made by cutting a slit into the stem of the host tree and putting a stem from another tree into the slit.
A hedge is a row of plants that have been planted very close together and trimmed into shape. Hedges can be used instead of fences to mark a boundary or to keep animals (or people) from getting over the fence.
Hedges are a very simple form of topiary. If you grow a lot of hedges together in a pattern, you can make a hedge maze.
A lawnmower is a tool for cutting the grass in your lawn. Lawnmowers come in several different types.
Lawnmowers can be ride-on like a small car, and you drive them around over your lawn. This kind is good for really, really big lawns.
Then there are push-mowers that don't have an engine and rely on you pushing them to spin a blade that cuts the grass. This kind are hard to use and are not very common anymore.
The most common is the kind in the picture, where you push it to make it go where you want, but the blade is powered by a petrol or electric motor.
Layering is a technique for propogating plants. It works best for plants that grow roots slowly, and plants that would naturally grow roots where a stem touches the ground.
To layer a plant, you usually take a stem and bend it down towards the ground, or into a pot, and hold it in place. You damage the stem where it touches the ground by cutting it part way through, adding some rooting hormone if you want. After some time, the plant will grow roots and shoots at the cut and it can be removed from the parent plant.
Layering can also be done in the air by cutting a stem part way through, adding the rooting hormone, and bundling up some damp potting soil or peat moss around the cut. The plant will grow roots into the bundle of soil, and then it can be removed from the parent plant. This works best with plants that aren't flexible enough to be bent to the ground.
Manure is any organic matter used for improving the soil. It can be pure animal dung (usually from farm animals), animal dung mixed with straw bedding, or pure vegetable matter.
Manure can be spread straight onto the fields where it is needed, or rotted down first into compost before being used.