All you really need to know, to understand drug dosages, is that doses are usually stated in milligrams (mg) or sometimes in micrograms (µg), relative to the the body weight of the patient (human or animal) expressed in kilograms (kg), and what those units mean. For those whose curiosity might lead them farther, there's a historical note at the bottom of this page.
The metric system of measurement is used worldwide in science and medicine. It is used in everyday life too, in most countries outside the U.S., but many Americans are unfamiliar with metric units. So what are milligrams, micrograms, and kilograms?
Weight in the metric system is measured in kilograms (kg). A kilogram is about 2.2 American pounds. Thus, a 10 kg dog weighs 22 pounds; a 20 kg dog weighs 44 pounds; and so on. Given any weight in kg, multiply by 2.2 to convert to pounds. To convert pounds to kilograms, divide by 2.2 (or multiply by 0.454, which is almost the same thing). Those equivalences between pounds and kg are not quite exact, but for our purposes they are close enough.
A gram (g) is a thousandth of a kilogram. That's not a very large quantity - about 1/28 of an ounce. A milligram (mg) is a thousandth of a gram - a very small quantity indeed. A microgram (µg) is a thousandth of a milligram, or a millionth of a gram - a really small quantity (that funny symbol 'µ' in 'µg' is the letter mu in the Greek alphabet, corresponding to 'm' in our alphabet). It's important not to mix up milligrams and micrograms, because if you did your dosage might be off by a factor of a thousand. Give a thousand times too little, and a drug would most likely have no effect; give a thousand times too much, and it might very well kill the patient.
Drug doses are usually expressed in relation to the weight of the patient. Thus, a dose rate of 10 mg/kg means giving 10 milligrams of drug for each kilogram of the patient's body weight. Divide by 2.2 to get the dose per pound of body weight, and you would get 4.54 mg per pound. For some drugs you might round that up to 5 mg/lb (really equivalent to 11 mg/kg). At 10 mg/kg a 20 kg (44 pound) animal would get 200 mg of drug. A 30 kg (66 pound) animal would get 300 mg; and so on.
The other thing you need to know, of course, is how often to give the stated dose. You may encounter some medical shorthand indicating the dosage frequency. For example, 'q' in front of a number means "every" so many hours or days. 'q8h' means every 8 hours; 'q3d' would mean every 3 days. Frequency can also be expressed as the number of times per 24 hour day that a drug should be given: 's.i.d' means once a day; 'b.i.d.' means twice a day; 't.i.d.' means three times a day; and so on. So you can see that 't.i.d' and 'q8h' mean nearly the same thing.
Thus if your veterinarian prescribes for your 20 kg (44 pound) dog a dose of 200 mg b.i.d., you know that the dose rate is 10 mg/kg and you are to give that dose twice daily, or roughly every 12 hours. Some people prefer to express that by saying that the dog is receiving 400 mg of drug daily, divided into two equal and equally spaced doses. The important thing is not to get confused. Always make sure you understand not only the quantity to be given, but also how often to give that quantity.
In the United States we use U.S. Customary Units deriving from the Anglo-Saxon Weights and Measures that evolved in mediaeval Britain. We are used to measuring distances in inches, feet, yards and miles, liquid volumes in ounces, pints, quarts and gallons, and weights in ounces, pounds and tons (although some of our units differ slightly - and annoyingly - from the corresponding British ones). In most of the rest of the world, traditional systems of measurement have largely been replaced by the metric system, invented about 300 years ago and established first in France, and then internationally, during and after the French Revolution. It is slightly ironic that although the American Revolution preceded the French one, which was inspired in part by our own success in throwing off monarchical shackles, the United States has now become the last significant holdout of the pre-revolutionary system of weights and measures.
Metric measurement is based on the decimal system. The basic unit of length is the meter, not too different in size from the English yard. It was originally defined as one ten millionth of the distance from the North Pole to the Equator along a particular meridian of longitude on the earth's surface. A more precise definition, in terms of the invariant speed of light in a vacuum, is now used. All other metric units are (or were originally) derived from this measurement of length. The decimal principle is used throughout. Thus a meter is divided into 10 decimeters, 100 centimeters, 1000 millimeters, one million micrometers (µm) and so on. The basic unit of volume is the liter, defined as one cubic decimeter, a cube 0.1 meter (10 centimeters) on a side. The basic unit of mass, as we have already seen, is the kilogram, originally defined as the weight of one liter of water at its temperature of maximum density (4° Celsius), or about 2.2 pounds. One milliliter (ml) - one thousandth of a liter - is a unit of volume equivalent to one cubic centimeter (cc). Going in the other direction, one thousand kilograms is one metric ton, and one thousand meters is one kilometer, about 0.6 of a mile. Put another way, one mile is 1609 meters, or 1.609 kilometers. So a U.S. speed limit of 60 miles per hour (mph) is nearly equivalent to a European (or Canadian!) one of 100 kilometers per hour (km/h).
One reason for the worldwide success of the metric system is that because it is decimal throughout, it makes calculations easy. Suppose you use U.S. measure to figure the weight of a water tank one yard on a side - a cubic yard of water. Well, a yard is 3 feet, so a cubic yard is 27 cubic feet, and a cubic foot of water weighs about 62.4 pounds, so 62.4 x 27 gives you about - well, about 1685 pounds. Most of us would need a calculator for that. Now suppose you use the metric system to figure the weight of a water tank one meter on a side - a cubic meter of water (one meter is a little bigger than one yard). A meter is 100 centimeters, so a cubic meter is one million cubic centimeters, and a cubic centimeter of water weighs one gram, so the tank weighs one million grams, or one thousand kilograms, or one metric ton. You can do that one in your head.
For the sake of convenience, 1/2 kg, 500 grams, is often called a "pound" in metric countries. It's about 10% bigger than an American pound. If you shop for, say, salami in a European delicatessen, you might order 100 grams (a bit less than 1/4 of a U.S. pound) or "half a pound" which would be 250 grams, 10% more than a half pound in the U.S. A metric "quarter pound" is 125 grams. Metric does not make daily life more difficult, though it might seem so if you're not familiar with it.
More information can be found on the home page of the U.S. Metric Association. Links from that site include the origin of the metric system, and a list of important dates in its history. "A Dictionary of Units" offers extensive and interesting information on the metric, British and American systems of units, and their conversions. The Wikipedia "History of Weights and Measures" offers further interesting information.