Ah, good. I saw this (or a similar) question a few days back but didn't have time to reply. Let me rectify that now.
As always, the usual disclaimer applies: countless books have been written on this over the past century; the below is just a quick overview of the key elements that governed relations between the Bolsheviks and peasantry. Notably it focuses on the political and economic sphere, omitting the cultural for the sake of conciseness. If you have any specific questions then let me know.
Pre-Revolution
The role of the Russian peasantry has been the subject of controversy and discussion since the late 19th C, underlining its importance. By 1914 peasants comprised some 90% of the Russian population and, in turn, some 90% of these households were organised around the traditional village commune (mir or obshchina). I give a quick overview of the obshchina here but the relevant point for us is the perceived inefficiency of traditional peasant farming.
How to reform agriculture was a key issue for the pre-revolution intelligentsia. The most concerted of these was Pytor Stolypin's 'wager on the strong': encouraging peasants to leave the commune to establish their own independent homesteads.
On the question of reform of agriculture, the Russian Social Democrats were very much in favour of modernisation. Going beyond Marx's contempt for the 'idiocy of rural life', the Russian Marxists had long been engaged in a polemical struggle with the Populists (Narodniki) over the possibility of building socialism in a peasant environment. The Social Democrats (Bolshevik and Menshevik) were of the opinion that capitalism was required first. Hence their understanding that the demise of the obshchina was not just desirable but inevitable.
Lenin's take on the issue had a slight twist. His Development of Capitalism in Russia argued that the entire Narodnik argument was moot as capitalism was already entrenched in Russia. He thus downplayed the robustness of traditional peasant structures (ie the commune) and emphasised stratification of the peasantry into rich (kulaks), middle (serednyaki) and poor (bednyaki).
In practical terms, none of this was yet relevant. The Social Democrats were almost entirely urban based and had little sustained contact with the peasantry. That would change.
Revolution & Civil War
One of the great ironies of the October Revolution is that - this, the archetypical proletarian revolution that delivered a Marxist party to power with the mission of creating a working class paradise – the big winners in 1917 were the peasantry. The Soviet Decree on Land, a grand stroke of Bolshevik opportunism/pragmatism (delete as appropriate), legitimised the parallel revolution in the countryside that had been building since February.
At a stroke the Revolution removed:
Individual land ownership. The nobles were driven from the countryside and Stolypin's independent farms reabsorbed into the obshchina. In both cases the village annexed their land.
Much state oversight. The Tsarist land captains were a thing of the past, replaced by weak Village Soviets that were easily controlled by the obshchina.
Much of the class tension in the villages. The 'Black Repartition' saw the enlarged village lands divided amongst its members, generally favouring the poorest peasants.
It took time for the significance of this to sink in. The Bolsheviks' initial line post-October maintained that the countryside was in the grip of class struggle between the rich and poor peasants. The duty of the Soviet regime was to support the latter in organising Committees of the Village Poor that would run the villages.
This 'class against class' approach wasn't a success. The Bolsheviks had little popularity in much of rural Russia, Soviet power in the villages was weak, rural class tension had diminished following the Black Repartition (being never as strong as Lenin assumed) and the peasant milieu remained surprisingly cohesive. All this rendered the Committees of Village Poor a, well, poor lever for influence and they were wound up in late 1918.
Yet Bolshevik policies were driven by the survival instincts as much as ideology: grain requisitioning, militarisation of transport and conscription continued. What perhaps saved them during the Civil War is that the Whites were even less popular – at least the Reds weren't threatening to reverse the Decree on Land. But by 1920-21 the country was at breaking point and intense peasant unrest (eg the Tambov Rebellion) forced the 'retreat' of the NEP.
NEP
The NEP was a fairly open acknowledgement that the Party was just not strong enough to continue coercing the peasantry. The restoration of market relations in the countryside was positioned as an alliance (smychka) between city and village. The Soviet state was still officially a workers’ state but with, Lenin was forced to concede, peasant characteristics. On the sliding scale of the Soviet class hierarchy, being of poor peasant background was second only to that of being working class, and allowed for relatively privileged access to education and other state services.
But this was not a happy alliance. The fundamental contradiction of modernising Bolsheviks and tradition peasant forms remained unresolved. Crucially, all Bolsheviks took it as a given that industrialisation of Russia was a priority and that the obshchina was a barrier to this. Agriculture therefore played a central role in the debates of the 1920s. On the right, Bukharin argued that a strong peasant market would provide the investment needed to fund industrialisation, notoriously telling the peasantry to "get rich!". This had too many echoes of Stolypin's wager for most Bolsheviks, however, who feared the re-emergence of a capitalist class of kulaks.
This link between the workers and peasantry finally sapped with the grain crises of 1927-28. This quickly escalated to full on collectivisation – breaking the power of the obshchina and corralling individual peasant farmers on to collective farms. This went hand-in-hand with dekulakisation as part of a broad offensive against the peasantry. Thus peasants were little more than sources of labour and cheap grain (Stalin's tribute, dan').
Wrap-up
In short, there was always a tension between the Bolsheviks and countryside, between thrusting modernisers and a peasantry that just wanted to be left alone. With that said, the interactions between the two were incredibly complicated and these were never two hermetically sealed worlds. The above is a just skimming the surface of the relationship.
Annoyingly, I discovered halfway through writing this that I'd done a quick comment from a similar angle awhile back. May be worth as a shorter read.
Sources
A collection of sources used above and good places to start.
Carr and Davies, Foundations of a Planned Economy 1
Davies et al, Economic Transformation of the Soviet Union
Figes, Peasant Russia
Koenker et al, Party, State and Society in the Russian Civil War
Mawdsley, The Russian Civil War
White, Lenin: Theory and Practice