The basic idea of putting a price on carbon is this: whenever you undertake an activity that results in the release of carbon dioxide (or another greenhouse gas with a similar effect), you are imposing costs on those who will suffer from global warming. Since people in other states and future generations are paying most of the cost, the emitter does not properly take it into account. The cost is ‘external’ to that person’s decision about how to behave. Making the cost ‘internal’ requires imposing a tax that increases the cost of the behaviour for the person undertaking it.
Actual carbon pricing schemes (whether tax based or cap-and-trade based) also need to choose between an approach based purely on internalizing the cost of carbon and one that also seeks to advance other goals. One motivation for the second option is political; it can be used to defuse opposition to carbon pricing within groups that are politically influential. Another motivation is the ethical notion that different people should pay more or less the same amount to combat climate change. Another motivation is the pure redistributive preference that exists within some political views and ideologies.
In the end, I don’t think any of these arguments is terribly strong. It makes sense to charge more to those who pollute more. Not only is that a matter of fairness, it is a matter of prudence. Knowing that the group is going to split the bill in a way that renders shares more even, a selfish diner will consume an above-average amount, counting on those who consume a lesser quantity to subsidize him. A revenue neutral carbon tax achieves the opposite: with heavy polluters paying dividends to those who are more restrained. Granting special treatment to politically influential groups also risks reducing the effectiveness of the carbon pricing scheme, partly because it becomes more worthwhile to try to game the political system, rather than cut emissions.
A carbon price should be a mechanism through which socially optimal behaviour is encouraged and the transition to a low-carbon society is advanced. It does things best when it is not also a vehicle for income redistribution on the basis of facts not relating to carbon, such as employment sector, family status, or income. Those things can best be addressed through other areas of taxation and policy, leaving carbon pricing focused on the achievement of environmental outcomes.