Basically, what happened after the February uprising in Petrograd was that Nicholas II and all of his potential heirs have been removed from power (or removed themselves, if you like). This left the Duma, the elected lower house of Russian parliament, as the sole legal authority. The executive, which, under the old regime, was appointed at the Emperor's discretion, and derived their authority from the Emperor's autocratic power, effectively lost all legitimacy. So the Duma appointed a provisional government from among its own members.

Under the 1907 electoral laws, the Duma was formed by indirect elections; seats were allocated according to complicated quotas which favoured relatively wealthy landowners and the richer strata of urban population (merchants, civil servants, businessmen, the upper professional class). The voice of a privileged voter was about 50 times heavier than the voice of a working-class voter, and about 125 times heavier than the voice of a rural commoner. As a result, the parties which represented the privileged voters had a disproportionately high representation. Essentially, the fourth Duma (elected in 1912) was divided between:

• The “right”, which were mostly conservative monarchists and Russian nationalists. They didn't have organized parliamentary parties and previously generally supported the Tsarist government agenda. After February 1917, they were sidelined.
• Centrists and liberals, mostly representing reform-minded privileged voters: their chief parties were the Constitutional Democrats (“kadets”), the more conservative Union of 17th October (“octobrists”), and the Progressives. The rather loose and ideologically diverse Centre Group can also be counted in this category. Generally, they wanted to reform the country as a constitutional monarchy along British lines.
• Socialists, which were divided in two doctrinal branches: narodniki and marxists. Their parliamentary parties were, respectively, the the Labour Group (“trudoviki”) and the S.-D. Workers' Party (consisting of Bolshevik and Menshevik factions). The Socialist-Revolutionary party (“esers”, radical narodniki), which enjoyed very considerable support among rural commoners, boycotted the 1912 elections, but, after the February revolution, a section of the Labour deputies reconstituted themselves as the S.-R. group. Socialists favoured a republican government with a greater political role for their constituents.
• Representatives of ethnic/religious minorities, mostly Poles and Moslems. Let's leave them aside for now.

The first Provisional government reflected this setup: it counted 5 kadets (including the chairman, Prince Lvov), 2 octobrists, a progressive, a member of the Centre Group, an independent liberal, and a labourist who then walked over to S.-R. (this was none other than Kerensky). Rodzianko, the chairman of the Duma, was also a kadet.

The Duma had to contend for power with the Petrograd Soviet (Council of workers' and soldiers' deputies). Unlike the former, whose legitimacy was based on their authority continuing from the Tsarist system, the latter enjoyed a wider support among the urban workers and military units quartered in Petrograd. These formed a more numerous class than the constituents of Duma parties, and, more importantly, they were an armed force. The Petrograd Soviet was at least nominally an elected assembly (one delegate for each thousand of workers or an army company), although the elections were probably not conducted in an orderly way. The original Executive committee was a self-appointed provisional group which grew out of a previous clandestine circle that coordinated labour protest (strikes etc.); it called the elections of the Soviet, and then handed over its leadership to a permanent Executive committee elected by the Soviet. The Executive committee was initially dominated by mensheviks, but also included some bolsheviks and esers. Several of its leaders, in particular Kerensky and Chkheidze, were also Duma members. Kerensky was both a minister of the Provisional government and the deputy chairman of the Executive committee. Councils similar to the Petrograd Soviet were subsequently formed all over Russia, a congress of their representatives was convened in June, and elected an All-Russia Executive Committee. Again, this was a body dominated by mensheviks and esers, with Chkheidze as chairman.

Given their lack of popularity among the wider populace and faced with rampant political unrest, the leaders of Duma liberals thought it wise to make a power-sharing agreement with the left, so the Provisional government was reconstituted in April 1917 as a broader coalition government which included two esers, two mensheviks and a labourist. However, during the spring and summer, the mass movement fell under an increased influence of bolsheviks (who campaigned for a government formed exclusively by Soviets — which they were gradually taking over) and anarchists (who were against any government). The street violence which they fomented, along with the beginning disintegration of the empire, disrupted the power balance and caused the Provisional government to be again reshuffled. This time it included 4 kadets, 2 radical democrats (a splinter group of the Progressives), 2 independent liberals, 5 esers, and 2 mensheviks. Kerensky became the chairman of the government. Subsequent events are largely explained by his political scheming and quest for absolute power, on one hand, and the bolshevization of the Soviets, on the other.