An adapter is a part which allows to place multiple stacks of parts beside a single stack. Although some adapters have “coupler” in their name they don't work as decouplers and are statically attached. Beyond the common structural parts, many fuel tanks can be used as adapter too.
All of these parts are Fuel Crossfeed Capable, meaning that when there is a fuel tank and an engine both connected to the adaptor on different sides, propellants, like liquid fuel and oxidizer, will flow through the adapter on the way to the engine.
The branches are unsupported and prone to bending or breaking if the weight and thrust are not properly distributed. Rockets incorporating an adapter are also likely to wobble, unless the stacks are opportunely bound together with struts.
The main uses of these adapters include connecting a small payload, such as a space probe, to a large launcher, or using several small engines on the bottom of a large fuel tank to increase efficiency. If a one-to-multiple adapter is on an upper stage which has engines underneath it, adding decouplers at the bottom of the engines and an upside-down one-to multiple adapter can allow for a large launcher underneath a powerful interplanetary vehicle. But only one of the decouplers are connected to the adapter below. This is very unstable and needs to be supported by struts.
Another, unconventional use of those parts is as a landing 'foot' when landing very large craft (such as bases). The stock landing legs aren't big or powerful, even though in version 0.22.0 they were re-modeled to include pistons for making a landed craft more stable. Using the strut parts as 'legs' and an adapter such as the Tri-Coupler as a 'foot', it can artificially make its own landing leg. Although, it will be mostly useless on sloped terrain, since it, unlike the landing legs, wouldn't have any pistons for keeping it from falling over on a slope.
Testing shows that the "tall" variety of adapters perform better aerodynamically than the "flat" variety, enough to offset the difference in weight. This means that when lifting off from a planet with an atmosphere, a larger delta-v is obtained when using the former (unless of course the part is protected from wind forces, for example using a fairing). The opposite is true when lifting off from a vacuum planet, due to the lower mass of the "flat" variety.