Was it similar to how the UK is administrated? Was there different laws and autonomous legislative bodies?
This is a naturally complicated question, as the USSR's structure changed several times throughout its history, so I will talk about its unification and a general evolution of the administrative system.
The Union's name came from the "soviets", workers' councils which were set up during the revolutionary period and which were councils assembled for workers to share opinions and coordinate. The unit of the soviet would become the key principle of Soviet administration from 1921 onwards.
There was an effective pyramid of soviets ranging from villages and sub-districts of cities, through provincial subregions, through provinces/oblasts/okrugs themselves, to the so-called "Supreme Soviets".
The USSR was an effective confederation of 15 "republics" or "nations", which united in the early 20s from socialist governments set up by the worker-appointed administrators and revolutionaries in each republic. They therefore maintained a republic-level assembly, which was the Supreme Soviet in each republic (e.g. the Supreme Soviet of the Russian Soviet Socialist Republic, descending to the Moscow City Soviet, then the Subdivision 21 Soviet, then even to soviets formed within housing blocks or streets).
We therefore see the soviet structure:
the Supreme Soviet of the Soviet Union, the national legislature which was elected by the population into two houses. Before its 1989 reforms, its great powers (to be discussed later) were intended to administer all aspects of the Soviet Union, but in fact it did little of the sort. Rather, it depended on a Presidium to do its lawmaking (to be discussed later).
the Supreme Soviet of [Republic], the legislature of each republic. Was technically endowed with a lot of power, but after 1925 exercised little power over anything other than strictly regional and administrative matters. On paper they resembled the regional parliaments and assemblies in the UK, but in reality had much less power.
the gorsovet, raysovet, selsovet and possovet , an elected body in each settlement with city status that handled local issues such as the provision of the militsiya (the civil and criminal police). Gorsovets were from cities, raysovets from regional sub-divisions, selsovets from rural areas and possovets from rural settlements. They had largely the same functions and oversaw local administration, including the sub-soviets formed in workplaces, apartment buildings, collective farms, etc, who reported to them and who elected their members and voiced their concerns there. They served a similar role to parish, borough and county councils in the UK.
Local soviets, although ostensibly under the control of the Supreme Soviets, were often controlled by appointed Party officials and trade unionists who ensured that Party control was heaviest at the lower levels.
Now we move onto how actual administration was done. While both the Supreme Soviets of the Union and of the Republic were technically imbued with great power, their only role was to meet several times a year, firstly to rubber-stamp legislation, but secondly to elect a Presidium, which was in fact the central governing unit of the USSR.
A Presidium is similar to a cabinet of ministers: a group of elected officials who have administrative power subject to approval of their legislature. In fact, they were not subject to approval in anything but name: the Supreme Soviets never rebelled against their opinions. Their role was to exercise the powers of the Supreme Soviets, ostensibly so that it didn't have to constantly be in session. This ensured an effectively dictatorial power structure, but dictatorship by committee which had a democratic front. The nomenklatura system maintained by Communist Party authorities (effectively a list of trustworthy and ideologically sound candidates) ensured that only those who would comply were ever elected to the Presidiums. The elections to the soviets were non-competitive, so only those appointed to stand by the Presidium were ever able to get into the soviets.
The Presidium of the USSR therefore were the power that enabled the Politburo, an elected group of Communist Party officials, to enact changes. The responsibilities of the Presidium included appointing the Procurator General (and therefore controlling the judicial system), changing the constitution, and any other legislative act whatsoever.
The administrative system was also accompanied by the equivalent Party structures: the Central Committee and the Congress of the Communist Party, which played an important role in the Stalinist and Khrushchev periods, and which operated on a similar principle - setting policy and enacting it.
To summarise, local administration was carried out by local appointed officials in city and regional soviets who supervised smaller groups of workers that were the inspiration for the system. Policy and law trickled down from the Supreme Soviets of the Republics and of the Union, which were "democratically" elected but in fact controlled by a Presidium which exercised all of the assembly's powers. These Presidiums obeyed Communist Party policy, set by the Central Committee and the Politburo, and turned it into law and administrative techniques.
There were also agencies such as the KGB and the military, which were under the control of the Party rather than the Presidiums. This is also why the USSR always had a Premier and a General Secretary rather than just one head of state.