Can the relationship between X and Y be graphed as a straight line?

In many experiments the relationship between X and Y is curved, making linear regression inappropriate. It rarely helps to transform the data to force the relationship to be linear. Better, use nonlinear curve fitting.

Are the data points independent?

Whether one point is above or below the line is a matter of chance, and does not influence whether another point is above or below the line.

Are the X and Y values intertwined?

If the value of X is used to calculate Y (or the value of Y is used to calculate X) then linear regression calculations are invalid. One example is a Scatchard plot, where the Y value (bound/free) is calculated in part from the X value (bound). Another example would be a graph of midterm exam scores (X) vs. total course grades(Y), since Y is in part computed from X.

Do you know the relative uncertainty of X and Y?

Ordinary linear regression assumes that you know the X values perfectly, and all the uncertainty is in Y. Deming regression assumes there is uncertainty in both variables. You have to specify the relative uncertainties, either by specifying that X and Y are equally uncertain or by entering the SD of each. If these values are incorrect, the Deming results won't be useful.