The Brandenburger were a special forces unit of the German army, initially under control of the Abwehr, the German army’s secret service, and from 1943 slowly moving to normal control channels. They started out as a relatively small, highly specialist unit, and by the end of the war had grown to the size of a regular field division. By that time, they had become more like British Commandos, or US Rangers. They were very active in the Aegean, and participated in the reconquest of Kos and Leros in October 1943, Operation Eisbaer (Polar Bear), which is best described in Anthony Rogers’ excellent book Churchill’s Folly.
A very good and succinct description is available in German at this link. This includes a list of commanders, sub-ordinations, and other information, including a discussion of the role of the Brandenburgers in the context of the laws of war.
In the context of Operation CRUSADER, the Brandenburger played a small role. I have been able to piece much of it together by the use of ULTRA intercepts and with the help of posters on the Axis History Forum. They had been requested to support the planned attack on Tobruk, possibly by a seaborne landing. A relatively small force was sent under Oberleutnant von Koehnen. This was from 13./Lehrregiment 800 Brandenburg z.b.V., and had been sent directly from Catania in Italy by plane on 14/15 November. The remainder of this company stayed in Italy, and was ready to be moved at the request of the Panzergruppe, although there are indications that this was not going to be possible before February 1942, maybe due to the transport situation following the destruction of the Beta/Duisburg convoy on 8/9 November 1941. The strength of this detachment was likely 1 Officer, 11 non-commissioned officers, and 70 men.
It appears that this detachment was then rushed to Benghazi to shore up the defenses there, and maybe split up on the way, with part of it remaining in Agedabia under the command of an men called Doehring, maybe a senior non-commissioned officer. On 29 November, von Koehnen was in Benghazi with 1 officer, an unknown number of non-commissioned officers, and 31 men. The remainder of the company was at the time in Italy, with a strength of 3 officers, 31 non-commissioned officers, 159 men, and with 17 lorries, 8 cars, and 3 tractors.
It is possible that another company (11./LR 800) arrived in Benghazi as part of Sonderverband 288 (see this older post).
During the main battle these units seem not to have been engaged. They were basically immobile, and had little or no heavy weapons. It appears that they conducted an operation on 22 January 42 during the counter-offensive. My guess is this would have been a small operation, maybe using English-speaking soldiers wearing Commonwealth uniforms to confuse the Commonwealth forces by giving wrong traffic directions (always a favourite) or impersonating officers to give false orders.
 lit. ‘men from Brandenburg’, the region outside Berlin
 lit. avoid/defend
 A ‘Halbkompanie’, half company – not a formation existing anywhere else in the Wehrmacht to my knowledge.
13th Company, Special Purpose Instruction Regiment 800