Adjacency works like this:
An improvement gives adjacencies according to the defined types (list at the bottom of the tooltip).
But it receives adjacencies according to its type, which is displayed at the top of its tooltip. Healing pools give adjacency bonusses for approval and influence, but it is only an approval improvement and thus gets no bonusses from a snuggler colony for example.
The Social Matrix for example is a manufacturing and a research improvement and therefore also accepts both adjacencies. However it does not get double adjacency bonusses if an improvement gives both (e.g. antimatter powerplant), it only gets one (the highest one, I guess).
In addition:
the bonus an improvement gets is independent on which kind of adjacency it gets. The bonus it gets only depends on the improvement itself. For example a slave mill (drengin tech) allways gets +5% mnufacturing, +5% research and +5% food, regardless whether it gets it from research or manufacturing adjacencies.
So for you power matrices this means you get +5% W/M/R from any population adjacency. But no others, since the Power Matrix only has the improvement type population.
For economic bonuses for example, should you put the economic capital on it or just a market or a bank? What about trade capital?
End of Mystikmind's quote
Depends on the bonus the improvement gives. Bank gives you +5% Wealth per level for example and a Financial capital gives 10% per level. So it is the financial capital. If you are playing with very low economic slider settings and have a lot of trade routes, which go to this colony, you might also want to consider trade improvements.
A manufacturing bonus, what is best on it? A factory? a manufacturing capital? or a duranthium plant?
End of Mystikmind's quote
Durantium plant. Especially early in the game. The other two give +5% manufacturing, durantium gives +1 manufacturing per level. They are amazing. They work independent of planetary production, i.e. they are not affected by slider, approval, coercion etc. And, since their bonus is flat, it gets multiplied with all other manufacturing improvements instead of being added to them.
What about when taking adjacency into account? say the bonus is on an outside tile that you cannot surround?
End of Mystikmind's quote
Example I wouldn't put an durantium refinery on an isolated island. You can get a solar power plant and a factory next to, which gives the same adjacency to the durantium refinery, but an additional two to the solar pp and factory each.
Early in the game I would place emphasis on maxing adjacency on promethion reserve, durantium refinery and thulium data archive (they are awsome!). But once you reach like 100+ population, it doesn't really matter that much anymore.