a semimartingale
A semimartingale is the most general process you can integrate against and still have a sensible stochastic calculus — it is exactly the right class of 'good integrators'. The motivation: stochastic integration needs an integrator whose increments are not too wild, but Brownian-type fluctuation (a local martingale part) and smooth-or-jumpy trend (a finite-variation part) are both allowed and both occur in applications (an SDE solution drifts and diffuses). Semimartingales are the processes built from exactly these two ingredients, and a deep theorem says they are precisely the integrators for which a good integral exists.
Definition: an adapted cadlag process X is a semimartingale if it admits a decomposition X_t = X_0 + M_t + A_t where M is a (cadlag) local martingale with M_0 = 0 and A is an adapted cadlag process of finite variation (its paths have finite total variation on every compact interval) with A_0 = 0. So a semimartingale is local martingale plus finite-variation drift. The decomposition is not unique in general (you can move a finite-variation local martingale between the parts), but if A is required to be PREDICTABLE then X is called a special semimartingale and the decomposition is unique — this canonical splitting is essentially the Doob-Meyer decomposition in disguise. Every local martingale, every adapted finite-variation process, every Levy process, every solution of an Ito SDE, and any smooth (C^2) function of these is again a semimartingale; the class is closed under the operations one needs.
Why it is the right notion: the Bichteler-Dellacherie theorem makes the heuristic precise — X is a semimartingale if and only if the elementary stochastic integral H -> integral H dX is continuous in an appropriate sense on simple predictable integrands (a no-free-lunch / good-integrator characterization). For semimartingales one has a full Ito formula, a quadratic variation [X], stochastic integrals against X, and stability under C^2 maps. The honest boundary: not every interesting process is a semimartingale. Fractional Brownian motion with Hurst index H != 1/2 is NOT a semimartingale (its paths have the wrong variation scaling), so the Ito theory does not apply to it directly — this is why rough-path theory and other extensions exist. And the local-martingale part being a LOCAL martingale (not a true one) is essential: insisting on a true martingale part would exclude many semimartingales.
The solution of an Ito SDE dX_t = b(X_t) dt + sigma(X_t) dB_t is a semimartingale: its martingale part is M_t = integral_0^t sigma(X_s) dB_s (a local martingale) and its finite-variation part is A_t = integral_0^t b(X_s) ds (the drift). Any C^2 function f(X_t) is again a semimartingale by Ito's formula, which is why the class is closed under the operations of stochastic calculus.
Drift plus diffusion: a diffusion SDE solution is the prototypical semimartingale, finite-variation part plus local-martingale part.
Not every process is a semimartingale: fractional Brownian motion with H != 1/2 fails the variation requirements, so Ito calculus does not apply to it — which is the motivation for rough-path theory.