Mushrooms are the fruiting bodies of fungi, so the question is really: are fungi plants? This is a vexing question and has led to fungi being placed in their own kingdom. At a cellular level, fungi and true plants share an important common characteristic. They all have a hard cell wall around the more flexible cell membrane. No animal has a cell wall and this indicates a closer relationship between plants and fungi than plants and animals or fungi and animals. However fungi are so different from plants in other ways that it is sensible to separate them into their own kingdom and answer the question in the negative. All plants, animals and fungi are eukaryotes, that is, they all have true nucleated cells. But they are different enough in other ways to separate them into different kingdoms. Fungi are neither plants nor animals and so mushrooms are not plants.
So what exactly are mushrooms? They are actually the equivalent of flowers in many ways as both are designed to propagate the species. However plant flowers produce seeds while mushrooms produce spores. Seeds are the product of sexual reproduction, where male and female gametes unite to form a zygote that contains the genetic material of both parents. Spores are asexually reproduced from the single parent that formed the mushroom. The purpose is to produce a hardy, drought or winter resistant form of the fungus that can be dispersed by wind or other mechanisms and thus ensure the survival and spread of the fungus to new habitats. But the spore is not the product of sexual reproduction as are the seeds of true plants.
The mushroom is a short-lived phenomenon but the fungus that made it can be quite long-lived. Fungi are made of hyphae, long strands of cells that grow by simple cell division. Below the surface of the soil, individual fungal hyphae grow through the soil, breaking down dead organic materials for energy and slowly moving by cell division and growth into new food sources. Eventually, when conditions turn unfavourable, either due to a lack of food sources, drought or the beginning of winter, the fungal hyphae grow together, entwine and produce a mushroom. This is the only visible sign of the presence of the fungus, whose individual hyphae may occupy hundreds of meters of soil under the surface.
The mushroom produces spores and then it dies. Sometimes the spores merely fall to the ground. In puffballs, they are squirted up into the air where they can be carried by the wind. Sometimes the mushrooms are injested and the tough little spores survive to fall with the feces onto a new patch of soil, where with the help of the fecal matter, they can sprout into new hyphae and start a new colony.
Most of the structures people call mushrooms are made by club fungi, the Basidiomycota, one of several phyla within the Fungi Kingdom. In this group the spores are attached to an umbrella-like structure called the basidium. The common button mushroom purchased at the local grocery story is a club fungus known by the generic name of Agaricus. There are other fungal fruiting bodies produced by the other phyla of fungi, including puffballs and the bracket fungi found on trees. All are fruiting bodies designed to produce spores and disperse the species. And none of them are plants.