The basic principle of numeric integration is you are calculating the area under the curve by splitting it up into smaller pieces and then adding up the results. There are several different methods to do that, and I recommend reading this Wikipedia page to get started:
Adaptive algorithms like the ones used in integral/integral2 track the error in the calculation, and continually subdivide the integration interval until the tolerance is met. Since they are able to automatically determine that some intervals require tiny pieces while others can use large pieces, they are able to handle "problematic" integrands but at the same time are very fast when the integrand is not problematic:
A common theme you will see in numerical solutions of ODEs and quadrature problems is balancing computation time with accuracy. You can always increase accuracy by using more intervals, or taking tinier steps, but it greatly increases the computation time. So adaptive algorithms that can automatically change the number of intervals/step size to meet the specified error tolerance provide a good balance.
Finally, note that integral2 has two different methods it uses depending on whether the integration interval is infinite or not:
