Plants are a kingdom of Eukaryotes (have cells divided by internal membranes and linear chromosomes). Fungi are a different kingdom, neither animals nor plants. Algae are no longer considered to be plants. Bacteria, including cyanobacteria (blue-green algae), aren't even Eukaryotes.
Non Flowering Plants
Angiosperms (flowering plants) comprise about 85% of known plant species. You might consider that non-flowering plants are bizarre. There are 11 other phyla (divisions) so you could argue that the angiosperms are weird.
The above phyla have alternation of generations; a generation with one set of chromosomes and a generation with two. They have mobile male sex cells that are similar to animal sperm.
Cycadophyta (cycads) look like palms but they're not. They have swimming male cells and take years to produce seeds.
Ginkgophyta has only one living member, the ginkgo or maidenhair, another living fossil.
Gnetophyta is a strange group with three living genera, in different families. Welwitchia is the only non-flowering plant sometimes considered to be a succulent. It has only two leaves, ever, which keep growing for its entire life becoming very tatty and frayed, estimated to be about 1000 – 2000 years.
These survive drought by storing water in their tissues, often seriously distorting plant anatomy. Usually part of the plant is fleshy to store water and reduce the surface area. The leaves are often reduced, short lived or completely absent. They often have spines to defend themselves or have disguise or camouflage. Although usually thought of as desert plants, many grow in mountains, rain forests (usually on trees – epiphytes) and salt marches (halophytes). Many angiosperm families have a few succulent members and several consist mostly or entirely of succulents.
The largest succulent family is Cactaceae (cacti), with branches reduced to small hairy patches (areoles). The leaves on these shoots are often modified to spines.
Many cacti are epiphytes, the best known being Schlumbergera (Christmas cactus).
Many have flattened stems, resembling leaves. True of many epiphytes and prickly pears.
Pereskias look like normal, leafy shrubs or trees but are actually cacti.
Blossfeldia considered the smallest cactus, with round, spineless stems. Often sets seed without opening its flowers as does Frailea, another diminutive genus.
Echinopsis tegeleriana is unique in having red flowers that open at night. It's a small cactus. Unlike most Echinopsis, it self pollinates, producing ridiculously huge, spiny fruits for such a small plant.
Some cacti produce flowers from especially spiny or hairy regions called cephalia at the top (Melocactus, Discocactus), side (Espostoa, Coleocephalocereus) or at the top and then growing through it to produce rings (Arrojadoa).
Pterocactus is the only genus of cacti with winged seeds. Also strange in having tuberous roots, a tendency to lose their stems in the winter and terminal flowers.
For some reason, many strange cacti come from Mexico.
Stenocereus eruca (creeping devil cactus) has stems that creep along the ground, heavily armed with backward pointing spines. They root as they go and die off at the old end.
Ariocarpus are small plants with large, leaf-like tubercles (extensions from the stem) and look more like Sempervivums (house leeks) than cacti.
Pelecyphora strombiliformis looks like a pine cone.
Obregonia denegrii looks like a globe artichoke.
Astrophytum has 6 species, all weird. Mostly have 5 or 8 ribs (ridges) and small felty patches (flecks) in addition to areoles. A. myriostigma and A. cohualense look nearly identical with usually 5 sided, spineless stems, but don't hybridize. A asterias is wider than high with 8 rounded ribs and tends to get mistaken for Lophophora (peyote cactus). A. caput-meducae is extremely weird with long tubercles, with an areole bearing small spines on the tip and a flower bearing areole about half way along. Sometimes there's more than one flower bearing areole on a tubercle or it produces more than one flower from an areole, both breaking normal cactus rules!
Leuchtenbergia principis, is a small cactus with long, leaf-like tubercles, looking more like an Aloe. The tubercles are tipped by areoles with long, papery spines that look as if they belong on an unrelated cactus. It branches dichotomiously (the stem forks), which is also unusual. The large yellow flowers are produced from the tubercle tips, unlike most large tubercaled cacti (e.g. Mammillaria).
Messembrianthemaceae is the second largest family of succulents, consisting mostly of leaf succulents (store water mostly in leaves). It includes many tiny plants (e.g. Lithops) that have fused pairs of leaves, usually gray or brown, resembling stones.
Crassulaceae - consists mostly of leaf succulents. Many species of Bryophyllum (sometimes considered a sub-genus of Kalanchoe) have plantlets (small plants that can drop off and grow) on their notched leaves or inflorescences.
Tylecodon consists of a truly bizarre selection of caudiciforms (storage organ on or below the ground), pachycauls (stem thin at the top and gradually thickening) and stem succulents, including T. reticulata, where the old inflorescences persist making it look as if the plant's covered in wire netting.
Euphorbia is a huge and ridiculously varied genus with flowers reduced to a single stamen (male) or a pedicel, ovary and stigma (female). These are arranged in structures called cyanthia which resemble bigger (but still small) flowers and often arranged in larger groups, sometimes with colorful bracts (leaves that are modified to look more like petals). About 500 of the species are succulents, often resembling cacti. A number of species have a main trunk with branches at the top, looking like a cross between a cactus and a monkey puzzle. A number of others (e.g. E. caput-meduca, E. flanaganii) have a thick main stem and long, sprawling branches. E. obesa looks much like Astrophytum asterias. E. piscidermis looks like a pine cone (or Peleciphora strombiliformis).
Fouquieria columnaria (boojum tree) has a cone shaped main stem (pointed end up) with twiggy branches at the top making it look like a giant, upside down parsnip. Some bend over to produce weird arches.
The Apocyanaceae - contains many weird succulents. Many species have flowers with the stigma, stamens etc. highly modified so these are sometimes referred to as the orchids of the dicots.
The variable genus Ceropegia usually have flowers with petals that remain joined at their tips, resulting in a strange lantern-like shape.
The stapeliads look like small cacti with 5 petaled flowers resembling those of Rafflesia, including the smell. Stapelia gigantea has flowers up to 40cm across!
Pachypodiums, have spiny, sometimes fleshy, stems and non-fleshy leaves. Their pretty periwinkle type flowers really look as if they belong on a different plant. The larger species (e.g. P. lameri) look like a cross between a palm and a cactus.
Boweia are bulbs which photosynthesize mainly on climbing inflorescences.
The Orchidaceae is one of the biggest plant families with species found in most parts of the world. They're famous for being expensive, difficult to grow and having weird flowers. The flowers have three sepals and three petals (like most monocots) but the top petal (the labellum) is highly modified. The pedicel and ovary are usually twisted so the labellum is at the bottom. The stamens and carpels are fused. Some species lack leaves, using their roots for photosynthesis or are parasitic.
A few plants catch and kill animals for nutrients. These all grow in very poor soils, usually swamps.
Sarracenia are pitcher plants that catch their prey in water filled pitchers on their leaves. They also have weird flowers.
Drosera (sundew) have leaves in a variety of shapes with blobs of sticky stuff on hairs. The prey gets caught in this and the leaf folds around it.
Dionaea muscipula (Venus fly trap) is probably the best know and most dramatic, with vicious looking traps on its leaves, which snap shut on its prey.
These grow on other plants or fungi. Some lack chlorophyll. Some can complete their life cycles without a host.
Probably the best know are the mistletoes (several genera of the order Santalales). These grow on a range of different hosts. There's even one that grows on other mistletoes!
Cuscuta (dodder) are sprawling plants, with thin stems, usually not green, with leaves reduced to scales.
Rafflesia arnoldii produces the world's largest flower, about 1m across, and it smells like rotting meat. It's a parasite on a jungle vine, existing as thread-like strands inside the host, the only visible part being the flower.
There are many bizarre plants which are hybrids, mutants, fasciated, variegated, chlorotic, chimeric or some combination thereof.
Many of these bizarre plants can be cultivated, some very easily, so you might like to have them in your home or garden.