Technology levels in Westeros are approximately similar to mid-to-late 14th century Europe, though there is some variation from one field to another.
Mid-to-late 15th century Europe is definitely the model for arms and armor. The average knight will wear mail armor reinforced wih plate at chest and limbs. Wealthier knights, especially of the Great Houses, might have fuller plate armor along the lines of that seen later in the early-to-mid 15th century. There is a great streak of pageantry in Westeros, and knights of the greater houses often have very decorated armor (especially for tournaments, though the greatest of the great might not flinch at wearing them in war). Silvering, gilding, enamel, and costly techniques to put color directly into the metal are all known and used.
For poorer knights, shirts and hauberks of mail with minimal or no plate reinforcement is common enough. In the North, mail is more prevalent than plate. In Dorne, the hot desert has led to the use of lighter armor, particularly scaled shirts. Unlike many of their contemporaries, the ironmen of the Iron Islands will wear mail armor for sea battles, believing that drowning in the course of battle merits a place in the Drowned God’s halls.
One area where Westeros is somewhat more advanced than Medieval Europe is in the area of healing. The maesters of the Citadel have hundreds of years of study and education before them, and while they do view health as being a matter of humors to some degree, they have some surprisingly advanced knowledge. Chief among these is that they have a very crude form of penicillin, by using bread molds on wounds to help deal with bacteria. They also know how to disinfect wounds with the use of alcohol, and appear to recognize the importance of keeping wounds and surgical instruments clean to prevent infection.
In shipping, galleys appear to be the most common vessels, especially for trade across the narrow sea. These galleys are quite advanced, and some of them appear to be as massive and well-constructed as late 15th and even early 16th century galleys are. The chief difference here is that cannons and gunpowder do not exist. Galleys in war are used to ram one another, to pull along side and affect boarding, and in some cases to use small engines of war to lob missiles and even pots of wildfire at one another.
"Greek Fire", a medieval weapon of war, does not seem to exist. But the aforementioned wildfire does, and it’s far more dangerous. Made by the Guild of Alchemists, known as the pyromancers, wildfire is a liquid substance which quickly ignites as a green, all-consuming flame that will burn on water and even consume metal over time. The substance seems to be partially magical in its formulation, and its secret is jealously guarded. It has been known to be used in naval engagements and in siege warfare.
Books are rare and expensive commodities, as the printing press does not exist. All books are copied by hand. A Great House might boast of a library containing a mere hundred books.
The Myrish have apparently invented simple telescopes, called far-eyes, and are known to make the finest lenses.