the drift and diffusion coefficients

In the SDE dX_t = b(X_t) dt + sigma(X_t) dB_t, the function b is the drift coefficient and the function sigma is the diffusion coefficient. Together they completely specify the local dynamics: b tells you where the process tends to move on average, and sigma tells you how violently it is shaken by noise. They answer the practical modelling question 'what data do I need to write down to define my diffusion?' — the answer is just these two functions of state (plus an initial condition).

Their meaning is infinitesimal and probabilistic. Conditioning on X_t = x and looking at a tiny later time, the expected displacement is E[X_{t+h} - X_t given X_t = x] = b(x) h + o(h), so b(x) is the instantaneous mean velocity; and the covariance of the displacement is E[(X_{t+h} - X_t)(X_{t+h} - X_t)^T given X_t = x] = a(x) h + o(h), where a(x) = sigma(x) sigma(x)^T is the diffusion matrix. So sigma controls the size and direction of the noise, but only through the symmetric nonnegative matrix a = sigma sigma^T; this a is what enters the generator and the Fokker-Planck equation. Different sigma giving the same a yield the same law, which is why weak solutions only see a.

These local coefficients are exactly the data that pass to the generator L f = sum_i b_i partial_i f + (1/2) sum_{i,j} a_ij partial_i partial_j f and to the Fokker-Planck equation for the density. An honest caveat: writing dX = b dt + sigma dB requires fixing the Ito convention; the same physical process in the Stratonovich convention has a drift that differs by the Ito-Stratonovich correction (1/2) sigma sigma', so 'the drift' is not coordinate-free unless you also state the convention. Also, when sigma depends on x (multiplicative noise) the second-order Ito term genuinely changes the dynamics versus the deterministic ODE with the same b.