This page contains information regarding Star Trek: Discovery, and thus may contain spoilers.
Location and size
Federation space was located approximately thirty thousand light years from the core of the galaxy, located within the Orion Arm, on the border of the Alpha and Beta Quadrants. (TNG: "The Nth Degree"; DIS: "Magic to Make the Sanest Man Go Mad"; VOY Season 7 production art)
In 2369, Cmdr. Sisko described the Federation to be made up of "over a hundred planets." In 2372, Cpt. Janeway said it "consists of over 150 different worlds." The following year, Cpt. Picard repeated this number and specified the Federation's territory to be spread across eight thousand light years. (DS9: "Battle Lines"; VOY: "Innocence"; Star Trek: First Contact)
During the Dominion War, the Female Changeling gave Weyoun the impression that once the Dominion won, that all Federation territories would fall under his jurisdiction, including Earth. Weyoun was later disappointed to learn that she later promised the Breen Confederacy the same terms, explaining that "I would promise the Breen the entire Alpha Quadrant if I thought it would help win this war." (DS9: "What You Leave Behind")
By 3189, approximately 120 years after the galaxy wide event known as "The Burn", which decimated warp faring civilization, the Federation was reduced to only 38 formal member worlds, from a high of 350. (DIS: "Die Trying")
Outside of Federation space
It maintained borders with many notable powers, including the Klingon Empire (Federation-Klingon border), the Romulan Star Empire (Romulan Neutral Zone), the Gorn Hegemony, the Tholian Assembly, the Cardassian Union (Federation-Cardassian border, DMZ), the Sheliak Corporate, the Talarians, and the Tzenkethi. (TOS: "The Trouble with Tribbles", "Balance of Terror", "Arena", "The Tholian Web"; TNG: "Chain of Command, Part I", "The Ensigns of Command", "Suddenly Human"; DS9: "The Adversary")
The Federation's Argus Array, a remote subspace telescope, was located at the very edge of Federation space near Cardassian space in 2370. (TNG: "The Nth Degree", "Parallels") The McAllister C-5 Nebula was located just across the Cardassian border, approximately seven light years from Federation space. (TNG: "Chain of Command, Part II")
According to the terms of the Klingon-Federation treaty, Klingons have "every right to free transit through Federation space" and they cannot be boarded or subjected to search without just cause. (TNG: "Gambit, Part II")
The Federation was generally referred to as an Alpha Quadrant power, particularly in Star Trek: Deep Space Nine, which presented the Dominion War as a fight for control of the Alpha Quadrant. Background sources, including the Star Trek: Star Charts, have long depicted the Federation as incorporating both Alpha and Beta Quadrant areas, an idea which was originally intended to rationalize Kirk's line in Star Trek II: The Wrath of Khan that the Enterprise was the only ship in the quadrant. (Star Trek Encyclopedia (3rd ed., p. 8)). Evidence of the Federation's multi-quadrant status long remained elusive, but a map seen in "Magic to Make the Sanest Man Go Mad" finally established otherwise.
According to the official Star Trek website, the Federation consisted of 120 members and 700 colonies as of 2258 in the alternate reality.  In a deleted scene from Star Trek Into Darkness, a region of space depicting Federation space was labeled with the title "Alpha Quadrant". 
According to Star Trek: Star Charts ("United Federation of Planets IV"), in 2378, the area of the Federation was eight thousand cubic light years and there were 183 members and 7,128 affiliates.