If you take a well-behaved physical system and perturb it a little bit, then you expect the total behavior or your system to be changed only a little bit. You can quantify this by saying that if your initial perturbation is $\delta$, then the final perturbation can never exceed $\gamma \delta$ for some constant $\gamma$. In many cases, such perturbations manifest themselves as small oscillations around the unperturbed orbit.
In a chaotic system, by definition, the deviation between the solutions for systems with minimally different initial conditions can become arbitrarily large. People first discovered this (at least a video on chaos theory I've seen in high school claimed so) when they tried to calculate the weather using some simple models. You quickly see that even very tiny changes in your initial conditions lead to completely different results for your forecast.
Now, as was pointed out in a comment, a tsunami is not triggered by atmospheric disturbances, but by underwater earthquakes. Maybe you think of "tornado". Here the answer would be yes and no. On the one hand, the very fact that there is such a thing as "tornado season" (i.e. right now in the US) and certain regions on earth with heightened frequency of tornadoes (tornado alley in the US) shows that it is not absolutely arbitrary when and where these things occur. The grand scheme, as you could call it, will not be affected by a butterfly's actions on the other side of the globe. But if you set out to calculate then when and where up to the millimeter and the millisecond, you would have to take into account all the tiny atmospheric disturbances.
Here is a mathematical definition from the Wikipedia article on the Butterfly effect:
A dynamical system with evolution map ft displays sensitive dependence on initial conditions if points arbitrarily close together become separate with increasing t at an exponential rate. The definition is not topological, but essentially metrical.