That is a question that can fill several books
I will try to give you an overview of the period in question, primarily concerned Northern Europe, particularly Scandinavia.
Around 300 AD there were two primary types of vessels: small vessels made of trunks (expanded log boats) and large clankbuilt rowboats. The small vessels, primarily fishing vessels, were built by hollowing out the trunk of a cut down tree. The remainder of the trunk would naturally tend to fold up and so the vessel was stiffened using ribs and beams.
The larger rowboats were clankbuilt vessels and could be quite large. The Nydam boat from around 310 - 20 AD (dated using dendrochronology) was approximately 24 m long and 3.5 m wide and could seat up to 28 oarsmen and possibly a few passengers. It was clankbuilt (the earliest known clankbuilt boat) and built using oak planks. Other boats using pine has also been found. These vessels was most likely warships but could simultaniously have functioned as trading vessels. The Nydam find also included an anchor.
The Romans was at this time still using their galleys and was using multible masts on their ships, as well as castles. These vessels regularly made trading trips as far as Friesland and possibly into the Southern Scandinavian area.
The sail is known from around 600 AD, in this area, but is not unthinkable that it has existed from around 450 AD or so. The first sailships was alot like the Nydam boat, but with a single central mast with a large square sail. The mast could be dismantled at sea, as the vessels was still primarily rowing vessels. Deckplanks became more common. These served as a cover to the boat's ballast and to provide a plane surface to load cargo upon (cargo was also placed beneath the deck). This is the first "vikingships."
Two types of seagoing vessels now developed: The millitary longboat and the the knarr. The longboat could be quite long, the Roskilde longboat (Roskilde 6) was 36 m long and one of the longest longships ever found. Most was somewhat shorter, however, about 25 - 30 m long and could seat around 70 - 80 warriors and their equipment. These boats was made for speed, long and narrow, and was not particular seaworthy, but was quite maneuvrable, as they were easily rowed.
The Knarr was shorter and wider, but could carry a massive load. Approcimately 50 tons was the maximum load of the later (9th - 10th century) ships. Their used the same basic building traditions as the longships. The primarily used their sails for propulsion, but early, and rare later, ships could be rowed by two or more sets of oars.
Ships was at this point still using a steering oar, fastened to the side of the boat at the rear stern (rarely both sterns). This system continued up to the 12-13th century. It is still highly debated how the ships were navigated. Most navigation occured using landmarks, but it is very likely that the norse had more advanced navigation techniques. They was most likely able to navigate using the stars. There has also been finds that point towards the use of a special disc. The discs had several lines that might serve a "sunline." It worked like a sunclock and had to be callibrated to the season, but it made it possible to keep a straight east-west course.
The cog appeared in the middle of the 12th century. They were flatbuttomed, and was ideal to serve along the coast of northern Germany and south-western Jutland. It was also in this area that they most likely originated.
This coastline is well known for its tides (and wildlife) ands so it was practical to have a ship that could stand on the seabed, fully loaded and not be damaged. The earliest ships was not much larger than the earlier trade vessels. They had steering oars and was initially clankbuilt, but this was quickly dropped, due to the strain on the buttom of the ship. They could load up to 120 tons of cargo, was 12 - 15 m long, and had a small hold beneath the deckplanks for ballast and cargo.
The cog was steadily developed and the rudder was moved to the stern almost from the begining. Shipbuilding was economized during the 13th century and sawcut planks were intruduced, easing massproduction of vessels. It was not until just after 1400 AD that ships was fitted with multiple masts and castles.
The millitary ships during this era was built using the same designs and it was not until the turn of the 15th century that warships began to increase in size. This was due to increased contact with the Mediterranean. It was also during this period that the first cannons found their way onto ships.
Barges was also used in this period and surfaced around the same time as the cog. They were built using the same basic construction techniques, but was wider and not as tall (early barges may have been in using in England around 1700 BC).
Exspanded logboats are known to have existed during the early viking age (793 - 950 AD) and possibly later. Rowboats (like Nydam, but smaller) was most likely in use in the viking age and later as well.
Smaller vessels was still in use after 1200 AD. Even as trade vessels. The cog could be compared to mordern bulk carriers, where smaller ships acted as coasters and small scale traders. Most of the immens seatrade in Southern Scandinavia was mostly done using these ships, which was very comparable to the viking knarr.
The compass was developed in the latter half of the 14th century and was a great leap forward in navigation.
Most shipping occured during the period from March to September/October. The seas are relative calm during this period and does not claim as many ships as winter would. During the winter months, the ships would be dragged onto land for safety and maintenance.
For further litterature I can recommend the
Ships and boats of the North series as well as books by Ole Crumlin-Pedersen. Litterature concerning the Bremen cog, the Oseberg ship, the Ladby and Sutton Hoo shipburials can also contain good information.
(I have always wanted to say that!)
