A dry cell is one specific type of galvanic cell (also called voltaic). Most batteries we use in our day to day lives (AAs,AAAs,etc.) are in fact dry cells. It is a perfect example of applied science.
Dry cells work on exactly the same principle as other galvanic cells (that of the redox reaction) but are designed for functionality by minimizing the weight and size of the unit.
In a normal cell, there is both the oxidant/reductant and their conjugates, along with an electrolyte (that allows ion movement to electrically balance the electron movement between the species) and an electron pathway. The electrolyte usually takes the form of an aqueous solution (water-based) and is therefore heavy and takes up a considerable volume. Along with the reductant and oxidant pairs, this leads to a rather large and cumbersome cell. It just wouldn't be as convenient carrying litres of water around with us in our discman or mobile phone, now would it?
A dry cell streamlines this by getting rid of everything unneccesary. The electrolyte is a moist paste which still allows ion movement but reduces the volume. The unutilised conjugate oxidant/reductant are removed and the casing itself forms part of the electron pathway.
In the most common type of dry cell battery, the well-known 'alkaline battery', the cathode (positive terminal) is composed of graphite, which serves as an electron pathway for the reduction half-reaction of Manganese Dioxide:
2 MnO2(aq) + 2e- + H2O(l) --------> Mn2O3(aq) + 2 OH-(aq)
A thin zinc cylinder serves as the anode (negative terminal) and it undergoes oxidation thusly:
Zn(s) + 2OH- ---------------> ZnO(aq) + 2H2O(aq) + 2e-
In the equations (s) stands for solid, (aq) for aqueous, an e is an electron and the superscript is the ionic charge.
This dry cell couple produces roughly 1.5 volts, and cells can also be linked in series to boost the voltage produced. The alkaline dry cell lasts much longer than the acid electrolyte equivalent since the zinc anode corrodes less rapidly under basic conditons.
Your standard double-A battery consists of one of these cells, however there are also many other combinations of oxidants and reductants that are used to produce different voltages in different batteries, such as the nickel hydride or the lithium ion batteries used in mobile phones and other similar devices.