a stochastic differential equation
A stochastic differential equation (SDE) is an equation for an evolving random process X_t that is driven partly by a smooth deterministic force and partly by noise. The standard one-dimensional form is dX_t = b(X_t) dt + sigma(X_t) dB_t, where B_t is a Brownian motion. Read it as: over an infinitesimal time dt the process moves a deterministic amount b(X_t) dt (the drift) plus a random amount sigma(X_t) dB_t (the diffusion). It is the model of choice for any system that obeys an ordinary differential equation on average but is constantly buffeted by random fluctuations: a stock price, a noisy electrical circuit, a particle in a fluid, a population subject to environmental randomness.
An SDE is NOT a differential equation in the classical sense, because Brownian paths are nowhere differentiable, so dB_t/dt does not exist. The meaning is given by the integral form X_t = X_0 + integral_0^t b(X_s) ds + integral_0^t sigma(X_s) dB_s, where the second integral is an Ito stochastic integral (constructed in the stochastic-integration field). The d-dimensional version takes X_t in R^d, a vector drift b: R^d -> R^d, a matrix diffusion sigma: R^d -> R^(d x m), and an m-dimensional Brownian motion B_t. A solution X_t is a process whose increments are described by b and sigma; calculus on it is governed by Ito's formula, in which the rule (dB_t)^2 = dt adds a second-order term that ordinary calculus does not have.
SDEs sit at the meeting point of analysis and probability: each well-posed SDE defines a diffusion process, a strong Markov process with continuous paths whose generator is a second-order elliptic operator, and through that operator it is tied to the heat-type PDEs (Kolmogorov backward, Fokker-Planck forward, Feynman-Kac). A crucial honesty: there are two genuinely different notions of solution. A strong solution is built on a given Brownian motion and is a function of that noise; a weak solution constructs the noise and the process together and only fixes the law. They need not coincide, and existence-and-uniqueness theorems must say which they mean. Also, the drift-diffusion split is convention-dependent: the same process can be written as an Ito SDE or a Stratonovich SDE with a different drift.
The simplest non-trivial SDE is geometric Brownian motion, dX_t = mu X_t dt + sigma X_t dB_t. Here the drift mu X_t and diffusion sigma X_t both scale with X_t, so the percentage change is what is driven by noise. Solving via Ito's formula on log X_t gives X_t = X_0 exp((mu - sigma^2/2) t + sigma B_t); note the -sigma^2/2 correction, which ordinary calculus would miss.
Geometric Brownian motion: the model behind Black-Scholes, and the cleanest illustration of the Ito correction.
An SDE is only meaningful as the integral equation; the differential notation dX = b dt + sigma dB is shorthand, since dB_t/dt does not exist. And you must say which integral convention (Ito or Stratonovich) you use — the answer changes with it.