Right from the start of the Bolshevik Revolution, the idea that units should have a political worker, politruk, was established. In the Russian Civil War every unit was meant to have one, though shortages of sufficiently politically sound men often prevented that. Political workers were there to ensure that the men were educated in Bolshevik theory and practice, to recruit members to the Party and to generally keep spirits high.
Some of these men became a Commissar, Komissar, with additional roles. They were to ensure that commanders were supporting Soviet power at all times. While in theory able to veto orders, in the normal course of events the commissar would leave the commander to run the strictly military affairs. In terms of battlefield tactics, they were generally too inexperienced in any case.
People tend to focus, as they did at the time, on the potential disloyalty of former Tsarist officers, voenspetsy, but in practice left-leaning commanders tended to be just as much of an issue, especially if Anarchist or Left-SR in tendency – many defecting because of disappointment at the path taken by the Bolsheviks.
The heavy handed commissars of later times should not be pushed forward to the RCW era. For example, when Stalin and Dzerzhinsky analysed the failure to hold Perm in late 1918 they criticised the ineffectiveness of the political education at the front. But even such staunch political warriors as they were did not suggest that commissars were failing to act appropriately in military situations – those days had not yet come. Commissars might sometimes over-rule commanders, and but most certainly could not shoot them (although they could, and did, report inappropriate behaviour to higher levels – and in extreme cases did arrest commanders for trial).
It was slightly different at the top. Every Army and Front, and Fleet for that matter, had a three person RMS (Revolutionary Military Soviet or Revvoyensoviets, RVS in the Russian): the commander and two political workers. Often called commissars, the RevKoms were technically different.
It is at this level that political interference in military matters could be seen, because the men appointed to such an important post were also men of some weight on the national scale. Stalin, for example, made a considerable nuisance of himself when acting on any RMS he was appointed to, basically taking over military control in some cases. This was not to the Red Army's advantage.
The usual Red Army. With maybe a big red star pinned to his chest.
The ranking systems introduced by the Soviets, at least those that make uniform books, don't seem to even have insignia for the ordinary unit Commissar.
Commissars in black leather jackets are a staple in books and internet, but they are quite hard to find evidence for. It seems to be based on contemporary Cheka agents (to whom they are unrelated) rather than any hard evidence – most Red commanders wore leather jackets, if they could find them, not just the political ones.
There isn't much need to represent Commissars in games, since they had little by way of battlefield role. Even if they were one of the sort to interfere with decision making, it can be represented by slightly amended morale rules (or in Red Actions!, by a different process of drawing commanders).
In situations where the Whites might have a regimental chief of staff, I use a Red Commissar (the Reds tending to be short on staff in general).
Trotsky's Directive to the Red Army on Commissars