Plants that survive drought by storing water in their tissues are called succulents. There's a huge variety of them. Some don't have any obviously fleshy parts (e.g. most species of Yucca). These are called borderline succulents. Others have fleshy roots, stems, leaves or some combination thereof and some bulbs are debatably succulent. Many plant families have some succulent members (e.g. the Apocynaceae). A number of families are composed more-or-less exclusively of succulents (e.g. the Crassulaceae). More about succulents.
The most famous more-or-less exclusively succulent family is the Cactaceae, better known as cacti. There are a large number of these are they're very varied but can easily be recognized by hairy patches called areoles, often with spines radiating from them, on their stems.
The Cactaceae is divided into four sub-families.
The Pereskioideae contains only one genus; Pereskia, consisting of ordinary looking, leafy trees and shrubs which grow in rain forests. These are borderline succulents at best but some do have tuberous roots.
The Mahuenioideae also has only one genus; Maihuenia, strange plants that form cushions of very spiny, fleshy stems with fleshy leaves.
The Opuntioideae includes the prickly pears and includes a number of genera. These have leaves but often they're small and short-lived. They also have nasty barbed hairs called glochids. Their stems are at least slightly fleshy and some have fleshy leaves or fleshy roots.
The Cactoideae is the largest sub-family. These have neither leaves nor glochids but some imitate leaves with flattened stems. Some have large tubercles (extensions from the stem) which can closely resemble leaves making the plant look more like an Aloe, Haworthia or Sempervivum than a cactus (e.g. Ariocarpus, Leuchtenbergia, Astrophytum caput-meducae). These usually have fleshy stems and sometimes fleshy roots. As well as the expected fleshy, spiny plants of dry places, a surprising number grow in rain forests on trees (plants that grow on trees are called epiphytes). These plants aren't very fleshy, often with thin, leaf-like stems. The best known of the these is Schlumberger, (Christmas cactus). There are also some hemiepiphytes, which climb up trees: Hylocereus and Selenicereus.
The outer layer of the stem of a cactus, the epidermis, is usually thick with a waxy layer called the cuticle, making it hard for water to evaporate through it.
Opening Stomata At Night
Stomata are small holes in plants, mostly on the leaves but they can be on the stems. They allow water to evaporate and gases, most notably oxygen and carbon dioxide to flow in and out. Most plants open their stomata during the day, absorb carbon dioxide and use it to make 3 carbon sugars using energy from light. Water loss can be reduced by opening the stomata at night. This uses something called CAM (Crassulacean acid metabolism) or C4 metabolism. In this the stomata are open at night and carbon dioxide is used to make malic acid. During the day, the malic acid is broken down and 3 carbon sugars made. Nearly all cacti do this, with the notable exception of Maihuenia.
Reducing Surface Area to Volume Ratio
If the plant has less surface area for its volume, it will have less area for water to evaporate from in proportion to the amount it can store. The larger something is, the smaller its surface area to volume ratio. Of course some cacti grow very large but there are also many small ones, just a few centimeters high and wide. The shape with the lowest surface area to volume ratio is the sphere but basically anything that's more or less fleshy looking will be better than thin stems and leaves. Also this goes well with having water storing tissue as this provides the extra fleshyness by adding water storing parenchyma cells and a thicker cortex. Most cacti have fleshy stems, some have fleshy leaves and some fleshy roots. The cactus that's closest to a perfect sphere is the very small and spineless Blossfeldia. Many are roughly ball shaped. Tephrocactus and Maihuneniopsis in the Opuntioideae have ball or egg shaped joints (stem sections, imagine a prickly pear with the flattened stems replaced with balls). Many Cactoideae, particularly the small ones, have round stems (globular cacti). Many more cacti have cylinder shaped stems. Usually these stems have ridges running from the top to the base (called ribs) or lumps (tubercles) to allow them to expand and contract depending on how much water they contain. A fair number of cacti stay close to the ground, reducing the amount of the stem that's exposed to the air. This may be achieved with a disc shaped stem with the growing tip the middle (e.g. Discocactus and Gymnocalycium ragonesii) or by having flattened stems with the growing tip on the side laying on the ground (e.g. Opuntia humifusa). Many other cacti have cylindrical stems laying on the ground (e.g. Stenocereus eruca and Echinocereus pentalophus).
Staying in the Shade
Plants need light to photosynthesize but keeping out of direct sun is useful to conserve water. Rainforests are shady so the Pereskias and epiphytes are in the shade most the time. Many smaller cacti and seedlings of larger cacti grow in the shade of rocks or other plants. Cacti usually shade themselves somewhat with their ribs, tubercles and spines. The most shady place is underground. Many succulents have underground storage organs and photosynthesize with short-lived top growth. The cacti that come closest to this are Grusonia pulchella and the genus Pterocactus, both in the Opuntioideae. Many more have fleshy roots (including those weird ones with leaf-like tubercles) but with stems that aren't lost on a regular basis: Blossfeldia, Ariocarpus, Lophophora, Peniocereus etc.
Reducing Evaporation From Flowers
Flowers are a major source of evaporation when they're open. Cacti mostly have large, showy flowers but they tend not to last long. Many cacti, particularly the larger ones (e.g. Cereus, Carnegia, Harrisia, Pachycereus), have flowers that open at night, which cuts down on evaporation. Some cacti have very small flowers, most notably the genera Mammillaria, Melocactus and Rhipsalis. Most have flowers that are somewhat tubular and trumpet shaped. A few have flowers that are very tube shaped, most notably Cleistocactus and Denmoza. Oroyas have flowers that are basically ball shaped. Many cacti have spines and/or hairs on the outside of the flower, which may provide a certain amount of shade. The very small cacti of Blossfeldia and Frailea have a strong tendency to produce seeds without actually opening their flowers. A number of cacti have their flowers in especially hairy and spiny regions called cephalia (e.g. Melocactus, Discocactus, Espostoa), which may serve to reduce evaporation from the flowers and buds. Most cacti produce their flowers from the areoles, which are usually positioned on the tops of ribs or tips of tubercles, but a few produce them from the tubercle axils (e.g. Ariocarpus and Mammillaria), possibly to give them more protection.
Some are better than others but most cacti are very good at storing water and can survive drought for an extended period. Some even grow in such inhospitable places as Death Valley and the Atacama Desert.
See also Cactuses growing in deserts