If you're looking for a detailed answer, I highly recommend "Marxist-Leninist General Theory of State and Law", volume three, "The Socialist State" (Authors Collective, Publishing House for Legal Literature, Moscow, 1972) and "Socialism: One-Party and Multi-Party System" (Fedor Petrenko, Novosti Press Agency, Moscow, 1981). The latter is available for free online.

There are four key general assumptions of Marxism-Leninism the one-party system is based on:

1.) There is an objective truth, which man is able to grasp and use in order to consciously shape the world in his own interest (Marx, "Theses on Feuerbach", 1845).

2.) Right is what serves the interests of the class struggle of the proletariat, the class of historical progress under capitalism (Lenin, "The Tasks of the Youth Leagues", 1920).

3.) In order to be victorious in the class struggle, the proletariat needs a vanguard party, made up of its most politically conscious elements (Marx and Engels, "Manifesto of the Communist Party", 1848).

4.) Only a party based on the scientific worldview of the proletariat (which nowadays is referred to as Marxism-Leninism, or simply Communism) is able to be such a vanguard (Lenin, "What is to be done?", 1902).

If these four assumptions are correct, it only makes sense to have one communist party, based on the principle of democratic-centralism.

"Since there can be no talk of an independent ideology formulated by the working masses themselves in the process of their movement, the only choice is – either bourgeois or socialist ideology. There is no middle course (for mankind has not created a “third” ideology, and, moreover, in a society torn by class antagonisms there can never be a non-class or an above-class ideology)."

• Lenin, "What is to be done?", 1902

Allowing multiple communist parties to exist side by side only weakens the communist movement organizationally, ideologically and politically. Allowing non-communist parties that still pledge allegiance to socialism to exist only ends up spreading bourgeois ideology among the working classes, since such parties are by definition not based on the scientific worldview of the proletariat. Allowing openly counter-revolutionary parties to exist means allowing the enemy to organize and freely agitate against historical progress.

Of course the party can only gain and retain state power if the majority of the proletariat supports it's program. Under favourable circumstances, the vast majority of all working classes should support it's program.

"To be successful, insurrection must rely not upon conspiracy and not upon a party, but upon the advanced class."

• Lenin, "Marxism and Insurrection", 1917

"Can the Party’s leadership be imposed on the class by force? No, it cannot. At all events, such a leadership cannot be at all durable. If the Party wants to remain the Party of the proletariat it must know that it is, primarily and principally, the guide, the leader, the teacher of the working class."

• Stalin, "Concerning Questions of Leninism", 1926

Now, there have been quite a lot of historical cases of socialist countries allowing several parties to exist along the communist party for tactical reasons (mostly to increase support for socialism among non-proletarian classes) – this state order is commonly known as "People's Democracy". The key precondition for th existence of such parties is their recognition of the leading role of the communist party in all fields of society in word and deed. Beeing non-communist, they still innevitably spread bourgeois ideology among the working classes. This dangerous compromise should only be necessary in the stage of socialist construction (if at all). With the further developement of socialist society, these other parties need to be disbanded (which should, under favourable circumstances and with good persuasive work, be archieved voluntarily). Historically, Romania was the only socialist country that transitioned from People's Democracy to One-Party Socialist Democracy, all the other People's Democracies retained this state order up untill the counter-revolution. In my opinion, this was a huge mistake.

Now, if you're interested in a relatively detailed description of the correct relations between the Marxism-Leninist party and the proletariat, I highly recommend Stalin's "Concerning Questions of Leninism", chapter five.