Usually nothing happens. The waves go over each other and act as if the other beam never intersected them.
This is called linear optics because light acts linearly and the phenomena is called [superposition](https://en.wikipedia.org/wiki/Superposition_principle).
But with high enough light intensity or suitable materials something can happen.
This something leads to [nonlinear optics](https://en.wikipedia.org/wiki/Nonlinear_optics).
There are many nonlinear phenomena that can happen. For example one photon can turn into two photons with lower energy, or two photons of lower energy can turn into one photon of higher energy. The optical properties of the medium can also change (for example refraction index that changes with total light intensity). Other more complex things can happen too.
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