"Meat" in this sense means, and has historically meant, what used to be called "flesh meat", which St. Thomas Aquinas described as "the flesh of animals that take their rest on the earth" [including birds, who rest in trees on the earth] (Summa Theologica, Second Part of the Second Part, Question 147, Article 8).
Exceptions to the rule of "no meat on Fridays" (and some other days) have historically been made for fish, and occasionally for other types of flesh—not that they were not considered meat (in the broad sense of "animal flesh" generally speaking), but that Catholics (either generally, or in particular places and circumstances) were allowed to eat them despite the fact that they were meat.
One famous example of such an exception is beaver. In the late 17th century, Francois de Laval, the first bishop of Quebec, was confronted by the fact that the First Nations peoples of Quebec, some of whom were Catholic, used beaver meat as a staple food. If beaver meat were considered "not flesh meat", some might argue that these Catholics were not keeping their fast days as required; but if it were considered "flesh meat" and thus prohibited, these people might starve, or fall gravely ill, for lack of other protein.
De Laval therefore submitted to his former theological colleagues at the Sorbonne the question of whether, in this case only, it would be in accord with Church teaching to make an exception for beaver meat. Based on the discussion from the Summa Theologica quoted above, the faculty responded that it would.
Note: The theologians did not answer (as some have told the story) that beaver was, or could be considered, not "flesh meat" (as if they had said that beaver was a kind of fish), but rather that it could be eaten by those in Bishop de Laval's diocese who relied on it without violating the rules of fasting and abstinence.
Similarly, in 2010, the archbishop of New Orleans, Louisiana, ruled that those in his diocese (and only those) could eat alligator meat without violating those rules.
Traditionally, the requirement to abstain from meat has sometimes also extended to animal products: dairy, eggs, and even soups and gravies made from meat. Aquinas seems to indicate that in his day this prohibition did not universally extend to all fasting days:
The eating of flesh meat is forbidden in every fast, while the Lenten fast lays a general prohibition even on eggs and milk foods. As to the use of the latter things in other fasts the custom varies among different people, and each person is bound to conform to that custom which is in vogue with those among whom he is dwelling.
(Summa Theologica, Second Part of the Second Part, Question 147, Article 8, ad 3)
Since the Second Vatican Council, even this requirement for Lent fasting and abstinence has been relaxed (as indicated in another answer). These products, of course, are not themselves considered meat themselves, but are forbidden because of their connection to animals.