What is the difference between relative-age dating and absolute-age dating?

The difference between relative dating and absolute dating is that relative dating is a method of sequencing events in the order in which they happened. Absolute dating is a method of estimating the age of a rock sample in years via radiometric techniques.
Short Answer:

The term relative dating is distinguished from absolute dating to make it clear that one does not get a specific estimate of the age of an object from relative dating, but one does get such an estimate of true age from absolute dating. There are several techniques employed in both sets of methods. Radiometric dating is one type of absolute dating.

Long Answer:

Sciences such as geology, Paleontology and archeology are very interested in identifying the age of objects found and these scientists sometimes use both relative dating or absolute dating to characterize the age of the objects they study.

Before radiometric dating (or other methods of absolute dating like counting tree rings) it was difficult to determine the actual age of an object. Radiometric dating, based on known rates of decay of radioactive isotopes in objects, allows a specific age of an object to be determined to some degree of accuracy.

Relative dating is a scientific process of evaluation used to determine the relative order of past events, but does not determine the absolute age of an object. The circumstances of the object may allow one to say that one object is older than another without being able to assign a particular age to the objects. Very often historical evidence is found in layers and older layers are further down that the top layers.

For example:

If an archaeologist is studying past civilizations, the archaeologist may be able to say that in a particular location the ruins of one civilization were found to have been built on another and so the layers unearthed in an excavation convey the sequence of historical occupations without revealing the actual dates. However, carbon dating is an absolute dating technique that can give an estimate of the actual age of an artifact and thus an estimate of the age of other objects in the same layer. Carbon dating is one example of radiometric dating.

Similarly, relative dating is done by paleontologists who find layers of fossils. By deducing which fossils are formed in the sequence of time, the periods when the particular fossilized entities existed can be arranged in order without the actual dates of when the fossils were laid down. The radiometric techniques that give absolute dating estimates are based on radioactive decay of elements such as uranium.

For geologists, it is similar. Looking at how rock formations are structured, a geologist may be able to say which rock was developed in which layer in a particular order but not be able to determine that actual geologic age of the layers. Geologists also have radiometric methods for absolute dating based on radioactive decay of certain elements.