Duma was Russian parliament and as such was official institution of the state. It predated revolution. It was controlled by political parties. At first by liberals that overthrew czar and then socialists.

Provisional government was the government of the Russian republic (established after first revolution that overthrew czar Nicolas II) and was subordinated to the Duma.

Soviets (literally councils in English) were effectively parallel parliament/government established by the people (mostly workers) during revolution and rival power of the Duma and the provisional government. They were organized locally (for example in factories) and then deputies would be send to the higher level Soviet (for example city). The most important Soviet during and after revolution was Leningrad Soviet, which served as an unofficial top Soviet. Bolsheviks gaining majority in the Soviets made them major power that could eventually defeat their opponents and gain control over country.

In the competition between Soviets and Duma/provisional government, Soviets eventually emerged victorious, mostly because they were able to control soldiers, railway, telegraph and factory workers. Basically they could control revolutionary masses while Duma and government could not.

As for actual political parties, there were liberals that actually organized coup that overthrew czar and started the first revolution (February Revolution). Two main liberal parties were Union of October 17 - otherwise known as Okrobtrists party and Constitutional Democratic Party otherwise known as Kadets. They organized anti government coup and forced czar Nicolas II to abdicate. Their idea was for Nicolas II to pass rule to his small son that would serve as a weak ruler of the constitutional monarchy similar to Great Britain. But Nicolas II refused and passed crown on to his brother grand duke Michael. However when liberals came to Michael, he told them that he will accept only if they can guarantee him safety, which they could not as coup got out of their hands and turned in to revolution with liberals having no control over masses on the streets and army.

Liberals were then forced to declare republic and created provisional government (until general elections), but were replaced in the Duma and the provisional government by Socialists who had majority in Soviets. Head of the provisional government became socialist Kerensky (hence "Kerensky government").

Main Socialist parties were Mensheviks, Socialists Revolutionary party called also Esers (basically socialist party that oriented itself on peasants rather then workers and performed political terror -political assassinations). Esers later split in to Left and Right Esers, former allied with Bolsheviks while later allied to other Socialists and liberals. Socialists were first to gain majority support in Soviets so they pushed liberals out of Duma and provisional government.

Then there were Bolsheviks, who before revolution split from socialist party of Mensheviks. They were more radical socialists. They were at first minor party that did not play any role in the revolution, but after Socialists took over Duma and provisional government, they begun gaining support in Soviets and eventually overthrew Kerensky, Socialists and their Duma and provisional government in the second revolution (October Revolution).

There were also various royalist parties and anarchists, plus parties of ethnic minorities (mostly of socialist kind) -for example Jewish socialists, but those did not play larger role.

During Kerensky/Socialist rule, there was failed attempt at counterrevolution organized by part of the army (mostly officers) and some political parties (originally Kerensky himself agreed to join but then he changed his mind fearing that he would be betrayed). Military force was send to take over Leningrad under general Kornilov and to disperse Soviets. Attempt failed as Kerensky flipped sides and most of the soldiers in Kerensky force refused to carry orders.

After Bolsheviks took power in October Revolution, general elections were held but as results were not flavoring Bolsheviks, Bolsheviks and Left Esers boycotted them which resulted in failed elections. After that civil war effectively started between Bolsheviks, Left Esers and few minor parties on one side and everybody else on the other.

At the beginning of the civil war Left Esers attempted coup against Bolsheviks, because they disagreed with their policy of making peace with Germany. Coup failed and Left Esers were purged.

Bolsheviks emerged victorious from the civil war and political struggle and united all political leftovers in to new Communist party.

[Edit:] One more important thing to understand is what made Bolsheviks and Left Esers disagree with other Socialists in Russia:

According to Marxist theory, development of human society comes in socio-economic stages:

Tribal society -> Slavery -> Feudalism -> Capitalism -> Socialism -> Communism

Russia at the beginning of the 20st ct. was in many ways still feudal society that just begun to adopt capitalism.

Most Russian socialists therefore believed, that Russia first have to became capitalists, before becoming socialist and then communist. And so socialists wanted to establish capitalist republic after revolution. Meanwhile Bolsheviks believed that it is possible for Russia to skip capitalism and go strait to socialism and wanted to establish socialist republic. That was the main point of disagreement.