What does this notation mean? Could anybody tell me what is this model? What does it mean?
$$\Delta Y = C + a(Y_{t-1} + bX_{t-1}) + c\Delta Y_{t-1} + d\Delta X_{t-1} + \varepsilon_t$$
Where $\Delta$ is the differencing operator, $a, b, c,$ and $d$ are coefficients, and $t$ is time.
 A: Technically the answer by @Aksakal is correct, but I suspect a different answer may be more useful given the context. Since cointegration was mentioned and the original model representation resembles very closely an equation from a vector error correction model (VECM) which is used for modelling cointegrated processes, I would guess the model in mind actually is VECM. 
A full bivariate VECM of autoregressive order $p$ would look as follows:
\begin{align}
 \Delta Y_t &= \alpha_1(Y_{t-1}+\beta X_{t-1}) + \gamma_{1,11} \Delta Y_{t-1} + \gamma_{1,12} \Delta X_{t-1} + \dotsb +  \\
            &\quad\ \gamma_{p,11} \Delta Y_{t-p} + \gamma_{p,12} \Delta X_{t-p} + \varepsilon_{1,t}  \\[10pt]
 \Delta X_t &= \alpha_2(Y_{t-1}+\beta X_{t-1}) + \gamma_{2,21} \Delta Y_{t-1} + \gamma_{2,22} \Delta X_{t-1} + \dotsb +  \\
            &\quad\ \gamma_{p,21} \Delta Y_{t-p} + \gamma_{p,22} \Delta X_{t-p} + \varepsilon_{2,t} 
\end{align}
The term $Y_{t-1}+\beta X_{t-1}$ is known as the error correction term, and vector $(1,\beta)$ is the cointegrating vector. The coefficients $\alpha_1$, $\alpha_2$ are known as loading factors. This model shows how a pair of cointegrated variables ($Y_t$,$X_t$) develop over time. The error correction term multiplied by the loading terms acts to hold $Y_t$ close to $X_t$, and vice versa (if the signs of $\alpha$s and $\beta$ are as expected).
I am not going to expand on the mechanics of the model further, but given the keywords you should be able to find relevant literature.
A: Yes, delta signs are differences, and time is $t$. Plugging the missing notation this is what you get:  
\begin{align}
Y_{t}-Y_{t-1} &= C + aY_{t-1} + bX_{t-1} + c( Y_{t-1}-Y_{t-2}) + d( X_{t-1}-X_{t-2})  \\
Y_{t} &= C + (1+a+c)Y_{t-1} + (b+d)X_{t-1} -cY_{t-2}  -dX_{t-2}
\end{align}
Which is ARX(2) autoregressive order 2 process with exogenous covariates:
$$Y_t = C + \phi_1 Y_{t-1} +\phi_2 Y_{t-2} + \beta_0 X_{t-1} + \beta_2 X_{t-2} + \varepsilon_t$$
Notice the lags on the exogenous variable ($X_{t-1},\ X_{t-2}$). They probably come with negative coefficients, which will make this model look like some kind of error correction model, as Richard Hardy suggested
