For an initial-value problem

$$y'=f(t,y),\qquad y(t_0)=y_0,$$

Euler's method advances by following the current tangent over a finite step $h$:

$$y_{n+1}=y_n+h f(t_n,y_n).$$

The method replaces the true curved solution over one step by a straight-line approximation.

For a sufficiently smooth problem, the error introduced by one step scales proportionally to $h^2$, while the accumulated error over a fixed time interval scales proportionally to $h$.

Reducing $h$ therefore improves the approximation at first order, at the cost of more steps.

Euler's method is simple enough to expose the essential structure of time stepping: evaluate the derivative from the current numerical state, use it to advance, then repeat.