I know the Wehrmacht was the Nazi German army, but I've read some books that refer to the SS as a "paramilitary group." Were they not actually officially sanctioned? What was the organization exactly?
The Wehrmacht was the official armed forces of the German state, including the air force (Luftwaffe) and navy (Kriegsmarine) as well as the army (Heer). The Schutzstaffel (SS) began very much as a paramilitary force - an armed force organised on military lines that isn't part of the official armed forces of the state. The SS began as a bodyguard force (Schutzstaffel = "Protection/security Force/squad", as part of the SA (the Sturmabteilung, "Storm detachment/division"), the main Nazi Party paramilitary force. At the time, many of the political parties in Germany had their own paramilitary forces. Himmler worked to make the SS independent from the SA, first achieving effective autonomy while still being officially part of the SA to finally crushing the SA and wiping out its leadership in the Night of Long Knives (in collaboration with the Gestapo, Göring's creation, which was handed over to Himmler to run a little over a month before the purge of the SA).
The SS was more than just the Waffen-SS. The SS had begun as Hitler's Praetorian Guard, but as it grew and replaced the SA, it expanded enormously, quadrupling in size from 50,000 to 200,000 in 1934. In September 1934, the SS-Verfügungstruppe (SS-VT, "SS Dispositional Troops") was formed, not just to continue the bodyguard function of the SS, but also capable of fighting on the battlefield. In 1938, Hitler clarified the relative roles of the SS-VT and the Wehrmacht: the SS-VT was not part of the Wehrmacht, and the Wehrmacht had no say in the control or activity of the SS-VT, which was up to Hitler and the Nazi Party. The Wehrmacht would help equip the SS-VT and help train them. As the war began, the Waffen-SS was formed from three armed branches of the SS. The largest was the SS-VT, consisting of three regiments, and Leibstandarte SS Adolf Hitler (one regiment, later to become an SS Panzergrenadier division and then an SS Panzer division), and the SS-Totenkopfverbände (SS-TV, "Death's Head Units") which ran the concentration camp system and contributed the majority of the manpower for the formation of SS Division Totenkopf which formed in October 1939 (and first saw combat in the Battle of France in May 1940, in Belgium and France, and committing war crimes aplenty right from the beginning).
While the Nazis were not in power, the SS was obviously paramilitary, and not part of the German armed forces, since it owed allegiance to the Nazi Party, not Germany. As the Nazi Party took over Germany, was the SS still paramilitary? State organisations can be paramilitary if they are not officially part of the armed forces. Such paramilitaries can include armed police forces and border protection forces whose chain of command lies outside that of the armed forces. Once the Waffen-SS went to war, they were operationally subordinate to the Wehrmacht and/or Heer, specially, to Oberkommando der Wehrmacht (OKW, the armed forces high command) and/or Oberkommando des Heeres (OKH, the army high command). However, their formal chain of command bypassed OKW and OKH, leading instead to SS-Hauptamt (SS Main Office) or SS-Führungshauptamt (SS Leadership Main Office, from August 1940 when it replaced the SS-HA), under Reichsführer-SS Heinrich Himmler. Some Waffen-SS units (about 25,000 troops in 1941), not in the battle zone, were under the operational control of Kommandostab Reichsführer-SS (Command Staff Reichsführer-SS), tasked with "rear area security", consisting largely of the murder of civilians and other war crimes and atrocities.
While the Nazi Party's and Hitler's attitude of "L'etat c'est moi" ("The state, it is I") partly obscures the difference between the armed forces of the German state and of the Nazi Party, the Waffen-SS remained a Nazi organisation, and therefore paramilitary. In the hypothetical and vanishingly-unlikely event of the Nazis allowing elections in, e.g., 1943, and losing and giving up power, the new German government would have, in principle, gained control of the Wehrmacht, while the Nazi Party would have retained control of the Waffen-SS.
The Waffen-SS was not the only Nazi paramilitary force that served a military function during the war. There was also the Nationalsozialistisches Fliegerkorps (NSFK, National Socialist Flyers Corps), which began as a pilot-training organisation before the Luftwaffe was formed. It continued to the end of WWII, carrying out air defence duties.