Bistability is a term for any dynamical system with two equilibria. That is, a system that can be at rest in two, and only, two different positions because these are dominant over their immediate neighbors. A bistable system will be at one of its stable equilibria most, though not necessarily all, of the time.
A two-dimensional model of state space can define bistability precisely. A bistable function has exactly two optima, which by definition the system is likely to move towards. In a graph, this looks like two troughs with a peak between them — assuming the system tends towards positions that are "lower" on the y-axis variable. These troughs do not necessarily have the same value; one can be deeper than the other. For the system to exhibit bistability, the troughs must only be sufficiently separate so that the system does not typically slide from the lesser optimum to the greater.
A relatively simple physical example of bistability is a light switch. If the switch is on, it stays on; if it's off, it stays off. Intermediate positions are temporary and tend towards these two optima. Other equilibria are rare or nonexistent.
In electronics, bistability describes a specific type of circuit called a flip-flop. A flip-flop is constructed for the purpose of holding one of two positions. These circuits change their value only according to a predictable input. The simplicity of this system allows it to serve as a building block for much more complicated apparatuses including computers.
Bistability can also apply to directly more complicated systems that do not conform precisely to this model. Other variables can have effects on the state of the system, and a sufficiently high degree of chaos can keep the system out of equilibrium. Modeling these systems as bistable can still be useful.
The difference between sleeping and waking in an organism, for example, can be understood as bistable. Awake and asleep are the two optima; intermediate conditions tend toward these positions. The physical activity correlated with these two states is extremely complicated, and there are many different ways to be awake, asleep, and somewhere in between. Nevertheless, the system has bistable tendencies. Once an organism begins waking up, this process usually triggers a large set of secondary signals that help it maintain wakefulness. Humans have found a variety of ways to complicate this bistability with drugs and indoor lighting, but bistability remains powerful even for them.