Adapted from a previous answer of mine:

The Communist Party of the Soviet Union owned a massive amount of property, especially real estate - (Communist Youth League buildings, Party Headquarters and Regional Offices, Party Resorts and the like), to say nothing of bank accounts and party media such as Pravda.

However, even before the August 1991 coup, the party was losing influence - and money. With the ending of the Party's legal monopoly on power and with the loosening of restrictions on political debate under glasnost, the party began hemorrhaging members. Over the course of 1990, the party's membership shrank from 19.2 million to 16.5 million: a loss of 2.7 million members (notably including Boris Yeltsin among their number). Resignations from the party accounted for some 1.8 million of these losses, and the rest from non-payment of party dues. The first six months of 1991 saw another million and a half or so leave, for a total from Jan 1990 to July 1991 of about some 4 million members, or a quarter of the total, leave the party.

Anyway, on to the Party and its dissolution in 1991. The Communist Party was structured in such a way that there was the Communist Party of the Soviet Union as the overall party for the USSR, and under it were republican communist parties for each of the 15 Soviet Socialist Republics (the Russian SFSR communist party was, ironically, the newest, only being formed in 1990 largely as a reaction against Gorbachev's reforms). The party was parallel with the state at pretty much all levels of government and society, with party committees and chairs at the republican, regional and local level, down to factory and collective farm cells.

The August 19-21, 1991 coup against Gorbachev was not organized by the Communist Party apparatus (but rather led by the KGB and Ministry of Defense), but many of the rank-and-file in the party at the very least stood to benefit from the coup, as it was largely a last-ditch attempt to halt Gorbachev's reforms and restore some degree of the monopoly of power that the CPSU held prior to glasnost'.

When the coup faltered, largely in the face of Yeltsin and the Russian republican government's resistance, the latter had their knives out for the party, and undertook what has been likened to a "counter-coup". On August 22 and 23, crowds in Moscow (with some degree of support by Yeltsin) attacked the KGB headquarters, famously removing the statue of "Iron" Felix Dzerzhinsky. Moscow city authorities (the city was governed by the democratic reformer Gavril Popov, who had been elected in 1990) diverted the crowds towards the Central Committee of the CPSU headquarters. With crowds surrounding the headquarters, and with rumors of party officials shredding documents (to hide their roles in the coup), Yeltsin and his associates forced Gorbachev to order all party members out of the office.

Members of the Russian parliament, in a televised meeting with Gorbachev later that day, demanded the full disbanding of the CPSU as a "criminal organization". Yeltsin signed a decree "temporarily" banning all party activity on Russian soil (the ban would be made permanent by Yeltsin's decree on November 6). The following day (August 24), Gorbachev formally resigned as CPSU General Secretary, urged the Central Committee to disband, and placed party property under the control and "protection" of local soviets (ie, local government). Yeltsin formally approved the takeover of party property the following day.

In almost all cases, effectively what was being done was transferring party property from control of one group of party members to other, now former, party members. Central Committee offices were placed under the control of the Moscow government. However, in many cases it turned out that the same people who had been in charge of the Party structure were given roles in the republican government: a large number of former Central Committee members were hired directly into Yeltsin's presidential administration, and at the local level many former party bosses simply moved into local governmental offices, to ultimately control the same properties.

In the case of Russia, while the central organs of the CPSU were dissolved and party property transferred to the state, local organs of the Communist Party were allowed to reorganize themselves in smaller parties (the Russian Party of Communists, All-Russian Communist Party (Bolshevik), etc.).

A case was taken by communists to the Russian constitutional court to rule on the legality of the decrees banning the CPSU and the confiscation of its property by the state (Gorbachev refused to participate in the trial). A number of Yeltsin's supporters filed a countersuit alleging that the CPSU had never been a legal party to begin with. After months of hearings, the court finally ruled in Dec 1992 that Yeltsin's decrees had been legal, but also that this only concerned the confiscation of party property and the disbanding central party organizations - local communist party organizations were legitimate. In February 1993 a number of these smaller organizations formed themselves into the Communist Party of the Russian Federation, which claimed to be a continuation of the RSFSR branch of the Communist Party. While it has been one of the largest Russian political parties in the post-Soviet period, its membership has numbered in the hundreds of thousands, not millions.

I should note that I've mostly talked about what happened in Russia, as it contained the bulk of party property and membership. Other republics largely followed Russia's path, although there were exceptions. In Lithuania most of the republican Communist Party members left the CPSU and formed a center-left party in 1989 that, through various twists and turns, became today's Social Democratic Party. In Uzbekistan and Turkmenistan the republican parties voted to change their names (to the People's Democratic Party of Uzbekistan and the Democratic Party of Turkmenistan), but largely continued on as-is, with the support of their respective Presidents.

As for the post-Soviet history of the Communist Party of the Russian Federation, which has been the largest (but far from only) Communist Party claiming descent from the Soviet Communist party: it has declined in importance since Putin became President (which is mostly outside the purview of this subreddit anyway). But it did have a fairly significant presence in Russian politics in the 1990s, and was widely viewed as a credible threat to Yeltsin. It had a strong base of support in regions in the so-called "Red Belt", which were areas (largely in southern Russia) that were heavily agricultural or had aging industries. The party came in first in the 1995 Duma elections, with 22 % of the overall party vote and 157 out of 450 seats. This Duma was particularly oppositional to Yeltsin, attempting to impeach him in 1999. In the 1999 Duma elections it received 24% of the party vote but only retained 113 seats. Communist Party member Gennady Seleznyov was Duma chairman from 1996 to 2003 (he abandoned other Communist deputies in a 2002 "portfolio putsch" to deliver a stronger legislative majority to Putin). From 1999 to 2002 the KPRF was in a tentative legislative coalition with the pro-Putin "Unity" party, with which it shared committee chairs, but it lost these with the 2002 upheaval (it was locked out of the new governing coalition).

The party, under Gennady Zyuganov, also posed a credible electoral challenge to Yeltsin himself in the 1996 Presidential Elections. Zyuganov received 32% in the first round, and 40% in the second against Yeltsin. Zyuganov ran again in 2000 but got 29%, and Putin won an outright majority in the first round. Since then, KPRF candidates (usually Zyuganov himself) contested all legislative and presidential elections but despite being perennial second-placers, received smaller and smaller shares of the vote.

During this time, it was really the only real "party" in a meaningful sense in Russian politics, as in one that had actual grassroots support (heavily based among pensioners) and anything like an ideological platform: many of the other parties in Russian politics in the 1990s tended to have much weaker bases of support, and were more often than not organized around single political figures.

With the rise of Putin, the Communist Party of the Russian Federation found itself a bit between a rock and a hard place: it was overall supportive of Putin (being an "official" opposition party in the Duma), while being critical of specific policies like changes to pensions. It wasn't really allowed to be a serious challenge to the regime, however, and the administration often coopted a lot of its language. While it maintained a sizeable following in the 2000s and beyond, it was much decreased in votes and members compared to the 1990s.

Sources:

Plokhy Serhii. The Last Empire: The Final Days of the Soviet Union

Remnick, David. Lenin's Tomb: The Last Days of the Soviet Empire

Karasik, Theodore. "Post-USSR: The Apparatchiki". Perspective Volume II, No 4 (March 1992)

Schmemann, Serge. "Yeltsin's Ban on Communists Upheld". New York Times. Dec. 1, 1992.

Erlanger, Steven. "Russian Court Weighs Communist Party's Legality"New York Times. July 8, 1992.

Jeff Berliner. "Yeltsin Bans Communist Party". UPI. Nov. 6, 1991

Not too well. After the short lived coup in August 1991, the CPSU or more precisely CP of the RSFSR was banned in the RSFSR in November.

After an internal struggle between hardliners and moderates which the latter won. They changed tactics from calling the post-soviet system illegitimate to the support of democracy and constitutionalism. The idea was to appeal the ban in court - and with some success, because the court allowed a basic party organisation. The problem at first was the party system with lots of other left parties.

The way out was was the election of Gennady Zyuganov as a moderate and nationalist, a follower of national bolshevism, kind of like Stalin. That made him an attractive choice even for conservative communists as well as moderate communists, basically any communist dissatisfied with Gorbachev. For a while, they didn't do that bad and were the biggest fraction in the Duma until 2003. Before that, in 1996, they had a genuine shot at the presidency, when the CP formed a coalition with nationalists and other communist organisations during the electoral campaign. In fact, it took Yeltsin two elections and a deal with the military to stay in office, rumours about fraud have never been entirely proven right or wrong. During the 1990s, they were even considered a major if not the biggest threat to democracy.

So in a way the Communists fared well until Putin. However, not only Putins tactics weakened the party. The CP was too conservative, became isolated internationally because it pondered towards a Slavic idea of national socialism. Communists on one side became more leftwing radical, on the other side they became involved with oligarchs like Chodorkowsky and others. According to Marxism-Leninism, those people were the enemy, so basically that maneuver weakened communists in Russia further. Moreover, the CP became complacent and satisfied - the party didn't really engage in voter and member acquisitions, it didn't take the role of an extra-parliamentary opposition or set up closed ties to unions - instead, party members gained publicity with anti-Semitic statements.

Another problem was and still is demographics. Of the more than 500.000 members in the late 90s, less than 1/5 of the party members hadn't been members of the CPSU. Aging was a major problem, because of that and other reasons, membership has declined to ~160.000 in 2017.

You might want to check out but keep in mind there might be updated editions:

March, Luke: The Communist party in post-soviet Russia (2002)

Barth Urban, Janet: Russia's Communists at the crossroads (1997)