A carpet is a soft, usually fluffy surface that is put down on floors. Carpet usually means something that covers the entire floor, but can sometimes mean a smaller rug. Sometimes people call carpet broadloom carpet or wall-to-wall carpet. Carpets make the floor feel soft and warm.
A carriage is a vehicle or a large wheeled container to carry people or things. In old times, it was pulled by horses. Today, people may use the word for a part of a train.
In modern times, horse drawn carriages are mostly used for occasional purposes, like tourism or weddings, not for daily travel.
Scientific name: daucus carota
Cascade means to fall over a steep surface or over an edge. Water is especially prone to cascading off of cliffs and forming waterfalls.
A case is something that goes around something else, for storage and protection. Expensive things that need to be kept safe are usually kept in cases. The picture is of a violin and its case, but it is common to see guitar cases and glasses cases too.
Case might also be a word for a situation or the state of things, as in "take some money in case you need it".