Definition of base SI units
Or "Where did they get that from, anyway?"

The International System of units is explained here. The following is merely a list of the seven primary international standards used to determine the base SI units. Note that with the exception of time/frequency, you are never ever going to see any of these standards. Ever.

Mass: measured in kilograms (kg);
Simply the mass of the international prototype. This prototype is a platinum-iridium mix (90%-10%) kept in a sealed vacuum at the International Bureau of Weights and Measures, Sèvres. It is kept in a vacuum because it was found to be gaining mass due to oxidisation effects.

Length: measured in metres (m);
Length has several definitions. The one most commonly bandied about is the 1983 definition, which is the distance light propagates in a vacuum in 1 / 299,792,458 seconds.
The previous definition from 1960 was "one metre is equal to 1,650,763.73 wavelengths in vacuum of the radiation corresponding to the transition between the levels 2p10 and 5d5 of the krypton-86 atom".
The original definition of the metre was a ten-millionth part of the earth's meridian passing through Paris. This was represented by the distance between two lines engraved on a platinum-iridium bar kept in the International Bureau of Weights and Measures.

Time: measured in seconds (s)
One second is the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-135 atom.
This standard is the only one accessible to the general public, as it is broadcast on various radio stations, including WWV, WWVB and Radio Rugby. This can be received on specialist clocks and suchlike pieces of equipment.
Frequency is not a base SI unit, but can be derived from time as 1 / s. It is measured in hertz.
What's that? Universal Time? Another node, methinks..

Current: measured in amperes (A);
One ampere is that constant current which, if maintained in two straight parallel conductors of infinite length and negligible cross-section, spaced one metre apart in vacuum, will produce a force of 2x10-7 N per one metre between them. That simple, eh?
Alternately, you can simply get two long conductors, put a large current through them, balance edge effects and other errors and then measure the resulting force between them. When it is 2x10-7N/m, you should have one ampere flowing through your conductors, provided your instruments are precise enough. Thanks to vuo again.

Temperature: measured in kelvin (K);
One kelvin is the fraction 1 / 273.16 of the thermodynamic temperature of the triple point of water.
Note that Fahrenheit is not an SI unit. Celsius is a derived SI unit, and is also measured in kelvins, but starting from 273.15 K.

Luminous Intensity: measured in candela (cd);
The luminous intensity in the perpendicular direction of a surface of 1 / 600,000 m2 of a black body at the temperature of freezing platinum under a pressure of 101,325 N/m2.
An alternate definiton, straight from the Bureau International des Poids et Mesures, is "the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540 × 1012 hertz and that has a radiant intensity in that direction of 1/683 watt per steradian."

Amount of substance: measured in moles (mol);
The amount of substance of a system which contains as many specified elementary particles as there are atoms in 0.012 kg of carbon-12.

Information from the IEE and the BIPM
Background from study notes
Correction to definition of ampere by vuo