Latin for "more" or "greater. Used to designate a more powerful version of a certain cartridge
. This extra velocity
is usually provided by a longer case which can hold more propellant
. More propellant will generate greater pressures which in turn will drive a projectile
The concept of Magnum is different from the concept of Super (like .45 Super, .38 Super or more recently .40 Super) in that Super versions are usually of the same external dimensions as that of the original cartridge. The case is usually beefed up a little in the critical internal areas where failure under high pressures are most common. These beefed up cases can withstand more pressure and thus allows for the use of greater amounts of propellant or the use of propellants which have a much higher burning rate. Be aware that as of this date there are no SAAMI or CIP standards for .40 S&W +P let alone .40 Super. These are either overpressured loads or just sneaky marketing techniques. Use of these rounds can lead to destruction of the firearm and injury to user.
Common magnums are the 44 Magnum, which is the more powerful brother of the 44 Special and the 357 Magnum which is the bigger brother of the 38 Special. Don't ask me why it is not called 38 Magnum as cartridge nomenclature does not make any sense, anybody who has had any subtantial experience with firearms will tell you that. It is also possible that S&W's head marketroid thought that "three five seven" sounded catchier than "thirty eight". Both the .38 special and .357 Magnum use a .357 inch bullet.
There also are magnum rifle cartridges, most recent at this date is the WSM or the Winchester Short Magnum. The WSM was magnumized by making it fatter rather than longer to allow its use in a rifle with a shorter action.