Until about 800 B.C. a variety of ships were used in the Aegean sea. However, they had not yet developed the basic forms of the classical age. Most ships were more or less symmetrical in design (similar to Viking ships, though much more primitive and less seaworthy), and the standard fighting technique was to board the enemy vessel.
Then, around 800 B.C., the ram was developed as a ship-to-ship weapon. Suddenly, speed and maneuverability became prime concerns.
The result of this development was the penteconter ("50-oared"), a fast galley propelled by 50 oars-men, 25 to a side, working in a single row. This was a pure warship, not a merchant-man. The length of a big penteconter was probably somewhere around 38 meters, which is quite near to the maximum one can achieve with the available materials. The beam was approximately 4 meters, which is the minimum width for a ship with two rowers that have to work abreast and want a decent lever for the oars. I have not found a tonnage for this ships. The top speed was estimated at 9.5 knots. A smaller version, the triaconter ("30-oared"), was also in use, with few differences except for the length of the ship.
The next important development is the bireme (Type I). Approximately around 700 B.C. somebody had the idea to use an outrigger. This was added to the open penteconter, and allowed two rows of oars. The lower row worked just as in the penteconter: The rowers were sitting towards the center of the ship, and the oars fulcrum was fixed on the rail. The rowers on the upper bench were sitting more to the outside (so they didn't interfere with the other row) and, to get a decent lever, their oars were fixed on the outrigger. Apparently the ship builders were willing to sacrifice some of the power of the lower row, because the beam of the bireme was only about 3 meters. Approximately 100 rowers were used, 25 per side in each of the two rows.
After getting the idea to use more rows of oars-men, the trireme was only a small step. According to Thukydides, the first triremes were build around 650 B.C in Corinth. Around 500 B.C. the trireme was the standard heavy warship for most Greek city states.
For the trireme the outrigger was a more integral part of the ship then for the older bireme. It basically was a wide rowing frame, sitting on the still very narrow hull. Now the two upper rows of oars were fixed to the outrigger, the lower one passed through the side of the hull proper. The ship also had a (partial) fighting deck above the rowers (later versions probably had a full deck).
The length of a trireme has been given as 35 meters, the beam as 3.5 meters (due to the outrigger the dimensions could be smaller than for the penteconter, and the ship could still pack more punch). The tonnage of the trireme has been estimated at approximately 40 metric tons (one third of which would be due to the 170 rowers). The top speed is usually estimated at 11.5 knots, although there are some speculations about ancient galleys actually entering the glide phase, which would defeat most of the wave resistance and allow speeds of up to 18 or 20 knots for very short bursts (wave resistance is the major factor in determining the top speed for a floating hull of a given lenght). There is convincing historical evidence that at least on one occasion a trireme with a crack crew managed to maintain 9 knots for 24 hours.
Now, the speed of the trireme is only 20 percent higher than for the penteconter, although it had three times as many rowers. This is due to the fact that the wave resistance is growing exponentially with the speed of the ship. However, a trireme has other advantages. It was much more maneuverable then the penteconter. It could accelerate from standstill to half speed in 8 seconds, and to top speed in approximately 30 seconds.
An interesting point is that the Greek trireme with 170 rowers was manned by only 14 to 20 marines. It had to rely on the ram - boarding maneuvers were seldom performed. The trireme was probably the most formidable ship ever designed for fighting with a ram. However, it had a number of drawbacks. It required a carefully trained and large crew. A single rower who couldn't maintain the stroke could cripple the ship for minutes. Therefore, only free men were used on triremes. If, in times of an emergency, slaves had to be used, they were freed before the combat. A whip or lash was not used (and wouldn't have worked).
One answer to the high cost and complexity was a new type of bireme (Type 2). It was developed by taking the lower row out of a trireme and shrinking the beam accordingly. With a length of 20 to 25 meters, and a beam of only 2.5 meters, the four rowers were working all abreast, and all oars would have their fulcrums on the outrigger. This bireme was smaller than the type 1 bireme described above, but it was much meaner, packing more power into a smaller, and faster ship.
These new biremes were cheaper than the trireme, and required only about 100 rowers. It was also easier to train the crew to row with only two rows of oars.
A completely different approach lead to the quadrireme, quinquereme and even larger ships. Now more people were used to move one oar, while the basic setup of a bireme or trireme was kept. This made the effort of a single rower less important. Also, these bigger ships could carry more marines (the Romans, who mostly used quinqueremes, packed 120 marines on the ships). They also made more stable platforms for catapults.
During the time of the Hellenistic diadochs this development was carried to the extreme, with large catamarans carrying up to 4000 rowers and lots of catapults. These ships proved to be quite effective for a time, but they had significant drawbacks. First, they were extremely expensive. Secondly, they were too slow, and hence of little strategic impact (you have to get your ships to the enemy before you can fight him). Also, the ancient seamen developed a wolf-pack tactic for the smaller ships, so that the catapults of the giants were not able to stop all of them from getting in a ram attack or a fire bomb on the giant ship - which sank a giant just as easy as a smaller ship. The battle of Actium, where Octacian's naval commander Aggrippa defeated the fleet of Antonius and Cleopatra, is an excellent (if late) example of this. Octavian and Cleopatra had about 220 ships, most of which were quinqueremes or larger, Aggrippa employed Liburnes, fast galleys with only one to three rows of oars.
At the same time the Diadochs fought for control of the eastern Mediterreanen, the Romans developed another approach to naval warfare during the Punic wars. They were not very capable seaman, but used large all-purpose warships (Quinqueremes are often quoted) and developed very effective boarding tactics (using a boarding bridge (the "corvus"), grapples, and similar devices). If this would have been effective for a long time is questionable. However, Rome fought few naval battles. After the empire was supreme in the Mediterranean, few warships were needed, and they were mostly relegated to fighting pirates.
Note: The Latin Bireme, Trieme, Quadrireme, Quinquereme correspond to the Greek Biere, Triere, Tetrere and Pentere. The terms are usually used interchangable today. There is some confusion about the use of the term Biere in early Greek - apparently, the old terms Triakonteros and Pentekonteros were originally used to describe both one and two decked ships, and the term Biere was only used later, as a translation of the Latin term Bireme.
Björn Landström: "Das Schiff" (German Edition, the Library of Congress lists the following data: Landstrom, Bjorn: Skeppet. English Title: The ship, an illustrated history, written and illustrated by Bjorn Landstrom. [Translated by Michael Phillips]).
This book has been published in 1961 and is probably still considered one of the best and most comprehensive sources on ancient and medieval ships. Most of Landström's reconstructions still stand, but new evidence has shed light in some of the darker areas. A more modern source, concentrating on the trireme and the larger ships of the Hellenistic monarchies is:
Bernhard Foley, Werner Soedel: "Ancient Oared Warships". The article appeared in Scientific American a couple of years ago - unfortunately I just copied the article and forgot to make notes of the exact volume.
Another important work I got only after writing the main body of the text is:
J.S. Morrison, J.F. Coates: "The Athenian Trireme: The History and Reconstruction of an Ancient Greek Warship", Cambridge University Press, 1986 (I own the German Edition, "Die Athenische Triere")
I added some remarks, especially about terminology and the Triaconter following their excellent presentation of the early source material. Their main work, the reconstruction of an actual trireme, is less accepted, and the final verdict is still out. However, their reconstruction of an ancient trireme, the Olympias is actually swimming (and their web site is a good starting point to explore the topic further).
Finally, I recently got "The Ancient Mariners" by Lionel Casson, which has some very interesting remarks on building methods and the development of the larger ancient galleys. Here is the full reference:
Lionel Casson: "The Ancient Mariners: Seafarers and Seafighters of the Mediterranean in Ancient Times (2nd ed.)", Princeton University Press, 1991
I also own a Time-Life book on ancient ships and seamanship. I have gleefully ignored most of the reconstructions they offer as implausible.