A simple example would be a pendulum. The DE is d²x/dt² = -kx and solving it gives x(t) = A sinωt + B cosωt. All very straightforward.
In real life the differential equations are far too complicated to solve analytically, but that's OK because we can point a big computer at them and solve them numerically.
But there are some differential equations that turn out to have an unexpected behaviour.
For these DEs we find the time evolution is very, very sensitive to the initial conditions. That is, suppose we specify the the state of the system at time zero then use a computer to solve for the variation with time.
If we make even the tiniest change to the initial conditions we find it makes a huge difference to the time evolution.
So suppose we take two systems that are almost identical i.e. differ by only a tiny amount, and compute their time evolution, then we find that after a short time the systems are completely different.
And this is what we mean by chaotic behaviour.
I need to go now. I won't be around tonight, but I'll be back tomorrow as usual.