The Technical Details: Determining Delta Values
Stable Isotopes
When we talk about the isotopic ratio in a sample, we talk about the delta value. Let's look at how a delta value is actually calculated:
 The first step in figuring out the δ^{13}C for a sample is to find the ratio of ^{13}C to ^{12}C within the sample.

Next compare (by dividing) this ratio to the ratio of ^{13}C to ^{12}C in a standard.
 There is a specific standard, with a known, unchanging ratio of ^{13}C to ^{12}C that all laboratories use in their comparison. For the stable carbon isotopes, this standard is a limestone (called Pee Dee Belemnite—or PDB) from South Carolina. Although PDB is no longer run as the standard, other carbonates (with a known, unchanging ^{13}C to ^{12}C ratio) are used and compared on the PDB scale. The carbonate standard is reacted with an acid to create gaseous CO_{2}, so that the sample and standard are both in the same phase.
 Often the sample and standard may have very similar ratios of the two stable isotopes, which will give you a value very close to (but not exactly) 1.

Many samples that actually have different ratios of ^{13}C to ^{12}C will give what seems like similar values (say, for example, 0.99 and 0.98). These samples do have different isotopic ratios, but this is hard to see when they only differ after the decimal point. To make this difference easier to see, 1 is subtracted from this value, and then this new calculation is multiplied by 1,000 to give the actual δ^{13}C of the sample.
 This makes it much easier to see the difference between two samples. For the ratios of 0.99 and 0.98, the delta values are 10‰ and 20‰ respectively.
The equation for this is:
Making the Values More “Friendly”
Even when comparing samples with ratios of ^{13}C to ^{12}C of 0.99 and 0.98, the delta notation is much easier. Well, when we look at ratios that atmospheric scientists actually study, it becomes infinitely easier to compare using delta notation–in fact it would be too difficult without!
Carbon Pool  ^{13}C  Actual ratio of ^{13}C to ^{12}C 

Ocean & Atmosphere  8‰  0.011142 
Terrestrial Biosphere  26‰  0.010945 
PDB Standard  0‰  0.011237 
Why Go Through This Much Work?
This seems like an awful lot of calculations when you can just look at differences among samples in their ^{13}C to ^{12}C ratios and ignore all of the calculation steps. The reason that it is conventional to compare to a standard (and then continue on to the next steps in order to get a more ‘friendly’ value) is so that it is easier to compare results both among isotope laboratories and within a single laboratory over a long time period. It is impossible to have an isotope ratio mass spectrometer that perfectly finds the ratio of ^{13}C to ^{12}C in a sample.
Isotope ratio mass spectrometers measure relative isotopic ratios much better than actual ratios. By comparing to a standard, the precision of the data values are much, much better since all values are relative to a given standard. For example, if the ratio for both the sample and standard are overestimated (or underestimated) by the same relative amount, then dividing the two values will account for this, making it possible to compare δ^{13}C among laboratories all across the world.
Radiocarbon Δ^{14}C
The formula for determining the Δ^{14}C of a sample is similar to δ^{13}C:
The difference is in the term F_{N[x]}, which is still a comparison of the sample to a standard. However, after this comparison, several other calculations occur to find F_{N[x]}.
 The ratio is corrected for “background” ^{14}C counts, where atoms or molecules that were accidentally and incorrectly identified as ^{14}C are no longer included.
 The ratio is additionally corrected for the small amount of radioactive decay between the time the sample was collected and the time it was measured, so that the Δ^{14}C at the time of collection rather than the time of analysis is reported.
 The final difference is that Δ^{14}C is normalized, where the effect of fractionation is removed. That is, we know from the ^{13}C measurements that, for example, when carbon dioxide is photosynthesized by plants, it fractionates, resulting in proportionately less ^{13}C in the plant. The same thing happens to ^{14}C, so plants have proportionately less ^{14}C than the atmosphere does. If we know how much ^{13}C fractionation occurs, we can calculate precisely how much ^{14}C fractionation there is. We then calculate how much ^{14}C would have been in the sample if it had not fractionated. This is the Δ^{14}C. Why go to all this trouble? The main reason is that for radiocarbon dating, scientists want to study how much ^{14}C has decayed, not how much has fractionated, and this normalization allows them to do just that. The second reason is that it makes it easier to understand the ^{14}C in the atmosphere – now when plants photosynthesize CO_{2}, the Δ^{14}C value in the atmosphere does not change. Of course, we can always reverse the calculations to discover the amount of ^{14}C without applying this normalization, and this is written as δ^{14}C.
For even more gory details, see: http://www.radiocarbon.org/Pubs/Stuiver/index.html