I'm aware of the very-basics like the General Secretary, the Politburo and Central Commitee, but could someone give me a simple rundown of the lower levels to the bottom? Also, I'm particularly interested in how the politicians at each of those levels were chosen?
The structure of CPSU was hierarchical. At the bottom there were the "primary party organizations", or a party cells. Normally one of them existed in each enterprise (factory, collective farm, university, or any other institution or organization). The minimum size was 3 party members, maximum could be very large. Next step was the area (район) organization ("areas" were geographical units: a small city, or several villages; larger cities consisted of several areas). Next level was the regional organization. A region (область) was also a geographical notion, of larger size. There were for example 24 regions in Ukraine. Next level was a republican organization, in all republics EXCEPT Russia. Russia did not have the republican organization. All this was controlled by the Central Committee, a small part of it was Politbureau. The chairman of Politbureau was sometimes called the General Secretary.
The system was called "democatic centralism". The members of the higher committees were elected by the committee one step lower. But all decisions made by the higher committee were mandatory for all lower levels. The inner working of the party was determined by the Charter (устав). According to this charter, the supreme authority was the Congress of CPSU whose delegates were elected by the lower level organizations. The Congress happened every few years, irregularly. The decision to call for a Congress was made on the top level. In practice by the General Secretaty. In practice the General Secretary had the power of dictator, but according to the charter, between the congresses, the supreme authority belonged to the Central Committee. It was Politbureau which decided to call a meeting of the Central Committee. Most of the time this was a dictatorship of the General Secretary who presided the Politbureau. Politbureau was elected by the Central Committee. In principle, the Central Committee had the authority to make changes in Politbureau, and this was used several times to change the composition of Politbureau, without the agreement of the General Secretary. For example Khrushchev, the general secretary was dismissed by the Central Committee. In practice the Central Committee was doing most of the time what Politbureau decided, and the same system was on the lower level: the decisions of the upper levels were mandatory for the lower levels.
There was a parallel "ordinary" structure of power: factory directors, university presidents, elected councils on the area, district, republic and all union level, as well as state departments (they were called ministries) etc. However no decision on any level could be made by administrations without the approval of the corresponding CPSU committee. And most decisions were made by CPSU committees first and then stamped by the organs of ordinary power. Most of the people in the ordinary administration were party members, and the decisions of their party committees were mandatory for them.
@Alex, while generally correct, presents a rather simplistic point of view. A doctrine of democratic centralism was indeed a cornerstone of power distribution in USSR. The important nuance is that to subordinate powers, according to the doctrine, the controlling organ must've appear united. That certainly gave the impression of dictatorship. In reality, far from it. Podgorny's removal illustrates very well that the Secretary General didn't consolidate enough power yet. As a side note, in a classic Brezhnev's Politburo an influence of people like Suslov, Grishin, Ustinov was comparable to Brezhnev's.
As for "ordinary power", it was also a bit more convoluted. A Director of an enterprise (akin to CEO), an University Rector (akin to President), etc, was appointed by a supervising organization (everything had a supervising organization), and was subject to approval by a corresponding Party organ. For example, a Director reporting to an USSR Minister had to be approved by Central Committee. There was exceptions: Directors of some extremely important Institutes were appointed directly by Politbureau.