What are they?
Basically, they are rythms that your body and mind seem to follow. Taking your birthday as a starting position, you plot the levels for each different cycle. From this you can (allegedly) make assumptions about a persons performance/ability etc.

History - where did they come from?
It all started in the 1890's when Dr Wilhelm Fliess did research on rythms in the human body involving 23 and 28 day cycles. He plotted the occurences of "fevers, susceptability to death, disease and childhood illness". He came to the conclusion that the human body was governed by 2 internal clocks. 1 on 23 days and the other on 28 days. The 23 day cycle appeared to regulate the physical wellbeing and the 28 day clock the emotional part. Interest picked up and many scientists (including good ol' Siggy Freud) became interested

How is it calculated?
Biorhythms are just are just cycles that start on your birthday and continue throughout your life. There are 3 cycles to consider (although I have seen a 4th talked about). The period of each cycle is measured in days. The emotion cycle lasts 28 days, the physical cycle lasts 23 days and the intellectual cycle 33 days. The possible 4th, intuition lasts 38 days. For each cycle, starting from day 0 (your birthday), you draw a sine wave with the appropriate period on a graph with days on the x axis and % on the y.
To calculate secondrary cycles, you take the sum of 2 primary cycles and plot this in the graph.

How do you interpret them?
Basically, you look at each cycle, if the cycle is near one of the extremities then you are approaching a critical point when the direction of change for that cycle reverses. On these days, you will feel either at the height of that cycle (eg at the height of the physical, you will feel generally more energetic) of low (the opposite). Days to watch out for are when the cycle is passing through the centre of the chart. This marks changes in the cycle and can lead to some funny behaviour from yourself.

How are they used in everyday life?
Japanese taxi firms have been using biorythms for years as a guide to when they should let drivers onto the road. It has been said that in 80% of all fatal road accidents, at least 1 of the drivers was on a critical day in his/her biorythm. People also use them as training guides, competing on peak days and resting on lows. The same applies for exams and and interviews.

To clear up the confusion from before - it seems that a lot of people also (incorrectly) spell it biorythm (without the rh). Corrections made and once again I needed a telling off.

Sources
www.Biorythm.co.jp