In a laser, the pumping mechanism excites an electron to a metastable state. This electron remains in the metastable state for a longer time and is de-excited to the ground state by emitting photons (this is called stimulated emission).
This process proceeds continuously to obtain sustained laser output. In the meantime, what happens to the photons emitted apart from moving between the mirrors?
In stimulated emission in a laser the emitted radiation has the same phase (and hence direction) as the incident radiation. The mirrors select some of those for regeneration back through the amplifier, where the process continues, and intense radiation builds up between the mirrors.
Some of the atoms will decay by spontaneous emission, and that radiation can go in any direction. This radiation is lost if it goes "sideways", and becomes noise if it is directed between the mirrors. Depending on the application, the noise may be important, or it may be of no consequence at all.
There's another mechanism that fits your description. The beam in the center between the two mirrors is small, but it expands due to diffraction as it propagates. A small proportion of the beam will miss the reflecting mirror, and is lost. This effect can be reduced to nearly-negligible by properly choosing curved mirrors, but it cannot be completely eliminated.
There are ways that photons can be lost without having them miss the mirrors. The primary mechanism is that typically one mirror is intentionally only partially reflecting so that light can get out and be useful. Some of the light in the laser scatters off of dust or anything and leaves the laser. Some of the light creates currents in the mirror, and the non-zero conductivity (in the case of metalized mirrors) causes ohmic losses.
