The Commons library briefing paper is a good starting point for the history, mechanism and issues if you are interested in some of the detail.
In short, the Barnett Formula is used to adjust the monies given to devolved governments and assemblies when UK public spending for services, that are devolved, is changed.
For example, health is largely devolved so, when the UK changes the overall health spending, say by increasing it by £1bn, each country will get a proportion of that increase based on a measure of their population. Scotland, for example, would get about £100mm because it has around 10% of the UK population.
Not all the monies for devolved powers come from this mechanism because not all of their funding is a proportion of UK spending. The Wikipedia article suggests that about 85% of the Scottish government's funding comes from Barnett Formula funding.
There are multiple criticisms from both sides of the debate.
The main payer criticism (for want of a better term) is a consequence of the fact that the formula only applies to changes in spending not level of spending. This assumes that the original split had some level of equity which isn't the case. As a consequence, the per capita spending in the smaller nations is significantly higher than in England. Even though England has some relatively poor regions.
The main receiver criticisms (again, I can't think of a better term) tend to fall into 2 camps. The first is that no effect, other than spending changes, is taken into account. So, for example, if tax is used to significantly benefit one nation, no adjustment is made. The canonical example is the 2012 Olympics where most of the benefit was aimed at London regeneration but the costs were applied across the UK.
In a similar vein, the Formula makes no allowance for need; just population. It would make sense for poorer regions/nations to deliberately get a higher per capita spend than wealthier regions but that isn't catered for in the Formula. Due to the nature of the Formula, the current spending advantage of the smaller countries is steadily reducing regardless of need.
As a minor digression, I'm not sure unionist vs nationalist really works to categorise the split. For example, English nationalists tend to be very anti-Barnett because it disadvantages poorer English regions with respect to the other nations. I'm not sure there are many pro-Barnett groups. It remains largely because agreeing an alternative will be politically explosive and there's not enough upside for anyone to justify it.