Descriptions of the different buildings in Star Kingdoms:

Residences - population is one of the most important things when building a kingdom. You can draft 1/10 of your civilians into soldiers every day, so the more population you have, the more soldiers you will be able to train. Each residence holds 50 civilians, which will increase to maximum of 60 when the population bonus is researched (some planet types also have advantages or disadvantages affecting their population). The good thing is that the civilians will pay taxes each hour, but the bad thing is that they require a little bit of power and can easilly be hurt by probings. Population will increase with 2,5% each hour as most.

Barracks - barracks hold military units (Soldiers, Troopers, Laser Troopers, Dragoons, Laser Dragoons, Tanks and Scientists), but NOT Tactical Fighters or civilians. Each barrack will be able to house 75 military units (tanks take up 2 barracks space each), so it’s good to always build enough barracks for all your military units and what you will train the next day. Because if you do not have enough barracks, the military will move into the residences, and kick out your innocent civilians. Look at "Status" - "Barracks Capacity" to see how many percent of your barracks that are currently used. Just like with power plants, it is good to always have slightly more barracks than needed, so try to keep barracks capacity below 100%.

Star Mines - Star Mines produce 120 money/hour each, which is increased when researching money bonus. You can use those buildings to increase your income, especially at the beginning.

Training Camps - the training camps will decrease the cost of training military units. For each % of your land used for training camps, the cost will decrease with 3%, until you have 10% training camps. After that, it won’t change any more. Training camps are good at the beginning, and I know that many of the best players have about 10% training camps all the game. But if you are smaller and get attacked a lot, it might be hard to keep that amount of training camps, and you will probably be able to use that land for other, better purposes (see "no training camps" in advanced strategies chapter).

Nuclear Power Plants - those buildings will produce 100 power/hour each, which will increase when researching power bonus and Energy Core. They will be able to store a maximum amount of 500 power each, and any excess power will be lost.

Fusion Power Plants - better than nuclear power plants. Will produce 140 power/hour + power bonus and energy core bonus, and store a maximum of 1,000 power each. Requires fusion technology research on 100% to be constructed, unless you play as Volcanic Inferno or Mystical Lands. They have those buildings available from the beginning.

Air Support Bays - can house 40 Tactical Fighters each, but requires Vesudian core & AI Technology research on 100% to be constructed. Without those buildings, you won’t be able to attack with your Tactical Fighters, and can't train any new ones either.

Probe Factories - manufacture probes. (you get between 0,5 - 1 probe for each factory you have every hour, which means that a person with 1000 factories get between 500 and 1000 probes/hour). Probes are really important, since they can be used for a lot of things. Without being able to probe a person to find out their military stats, you’ll probably not be successful when attacking them either. Besides, if you have a lot of probes, the possibility that other persons will be successful in probing you decrease too. You can also use them for robbing persons, which gives you 10x the number of returning probes amount of money (or a max. of 10% of the probed kingdom’s savings), if the person you probe has that much.

What buildings to construct is your own choice. However, without enough barracks, your military will kick out your population, and without powerplants, you will go powerless, so those two buildings should always have the highest priority, unless you use an advanced playing strategy where any of those buildings are useless (see Advanced Strategies chapter later on in the guide).
You can look at "Stats" in the menu to see if you have enough barracks for your recent military and what you have in training. If you don't have that, the barracks capacity value is above 100%. If you have too few powerplants, you will notice it because you have a very low power income (or lose power every hour), if you look at your kingdom status. It is good to always be a little bit ahead in both barracks and powerplants.