A magnetosphere is a region that contains a planetary magnetic field, separating it from the local magnetic environment; in our case, the solar wind. The magnetic field usually emanates from some sort of geomagnetism (ie, internal planetary currents). The solar wind flow compresses the magnetosphere on the day-side and confines it to a magnetospheric tail on the night-side (which can be enormous), making the overall shape of that of an elongated raindrop.
The state of solar activity (solar cycle and coronal mass ejections) can have a profound effect upon the state on the magnetosphere, termed the 'space weather'. Satellites, global communications and power grids can be affected. Recent results suggest that extreme solar events can also affect global weather.
The Earth's magnetic dipole (or moment) points southward, making the field northward as it crosses the equatorial plane. The opposite is the case for Jupiter. This means that at the poles the field lines point into (or out of) the planet, allowing particles to penetrate into the ionosphere and atmosphere, possibly causing aurora.
Usually a magnetosphere is full of plasma (the fourth state of matter, ionised gas) which orbits the planet following the laws of plasma motion and magnetohydrodynamics (the study of the macroscopic behaviour of the plasma as a magnetised fluid).
Due to the electric and magnetic forces acting upon it, plasma can be trapped in so called radiation belts, examples being the Van-Allen belts at Earth, and the equatorial current sheet at Jupiter.
This plasma usually emanates from the planetary ionosphere (Earth), from a moon (Io: Jupiter, Titan: Saturn), or from the material contained within the solar wind (Earth).
Two major types of global flow are observed within magnetospheres:
Corotation - The planet imparts angular momentum upon the magnetosphere by both magnetic tension forces and field-aligned currents.
The Dungey Cycle - The solar wind attaches itself to the dayside magnetopause (via magnetic reconnection), and pulls the system down wind in all day-night planes, with a return flow being seen through the inner magnetosphere, in order to obey flux-conservation.
In reality, some sort of superposition of the two above flows is observed.
Spacecraft Studying Magnetospheres
Many spacecraft have been launched to study the Earth's magnetosphere, the most recent and notable being the Cluster II mission. Launched recently, Cluster II comprises a set of four identical craft capable of distinguishing between temporal and spatial variability. Recent missions to the outer planets are the Galileo craft which has been in orbit around Jupiter since 1995, and the Cassini mission to Saturn, due to arrive in 2004.
The NASA website has some good information and graphics on magnetospheres.