There are several different processes that can lead to the escape of a planetary atmosphere. In some cases this can be a very important process; for example, both Venus and Mars have probably lost much of their water.
Thermal escape mechanisms
In normal thermal escape (sometimes known as Jeans escape), gases generally escape very slowly. A gas is made up of atoms or molecules with a wide range of velocities. If the fastest particles in an atmosphere reach escape velocity, then they slowly escape into space. The more massive a gas molecule is, the lower its average speed at a given temperature, meaning it is less likely to escape. This is why hydrogen escapes from a given atmosphere more easily than carbon dioxide. Also, if the planet has a higher mass, the escape velocity is faster and fewer particles will escape. This is why the gas giant planets are able to have significant amounts of hydrogen and helium, while they escape on Earth. The distance to the Sun also plays a part; a close planet has a hotter atmosphere, which generally leads to a faster range of velocities, and more chance of escape. This helps Titan, which is small compared to Earth but further from the Sun, keep its atmosphere.
However, while it has not been observed, it is theorized that an atmosphere with a high enough pressure and temperature can undergo a 'blow-off'. In this situation molecules basically just flow off into space. Here it is possible to lose heavier molecules than would not normally be lost. This might possibly be a way for Venus to lose its water early in its history. It is closer to the Sun than the Earth so that early in its history its oceans probably boiled off. Assuming it started with a similar amount of water to Earth, evaporating all its water would lead to an atmosphere with a pressure of 270 bars (the surface of the Earth is at 1 bar) composed almost entirely of super-heated steam.
Solar wind escape mechanisms
There are a wide range of ways that the solar wind can lead to the loss of atmospheric particles. Generally, they involve charging the particle that will be lost, which leads to 'pick-up' by the charged solar wind. These mechanisms usually follow the same pattern as the Jeans escape described above, in that they are more likely to erode lighter atoms from lighter planets. However, in this case a magnetosphere helps protect against loss. It deflects the solar wind, and prevents its ions and magnetic field from carrying away atoms.
On planets without a magnetosphere, some combination of solar wind mechanisms very often dominate atmospheric escape. Both Venus and Mars are currently losing their water this way. First, the water is dissociated into hydrogen and oxygen by ultraviolet light from the Sun, and then the light hydrogen is pulled away in the solar wind. In fact, hydrogen from Venus has been detected at Earth.
The impact of a large meteroid can lead to the loss of atmosphere. If a collision is energetic enough, it is possible for ejecta, including atmospheric molecules, to reach escape velocity. Just one impact such as the Chicxulub event does not lead to a significant loss, but the terrestrial planets went through enough impacts when they were forming for this to matter.
This is perhaps more of a loss than an escape, because this is when molecules solidify out of the atmosphere onto the surface. This happens on Earth in glaciers or when carbon is lost to sediments. The dry ice caps on Mars are also an example of this process.