Nothing fancy, much the same way things are built today. If needed construct a framework and then simply add one piece onto another until you have what you want.
With the discovery of other dimensions, or more specifically with the discovery of how to access them, they can be utilized for construction purposes. A porthole into another dimension is created and built (discretely) in a door frame or other opening. What is then built beyond that opening, while still apart of the rest of the construction, is in a different plane of existence. The end results are limitless.
A small ship that is vast on the inside, able to carry thousands of people. Pockets in clothing that can hold as much as you care to put into them. Secret storage areas that cannot be found through conventional means. Robots able to pull guns out of nowhere because they have dimensional pockets they keep near them at all times.
One word of warning about extra dimensional construction, however, is to keep in mind what dimension it is you are building in. Is it inhabited? If so will the beings within be benevolent and simply take care of what you place in their care, or will they be destructive and looking for any door out. Is the dimension going to cause extra ware on what you place within it? Does the dimension have wind, will your constructions drift? Each question has different answers depending on exactly which one happens to be chanced upon using.
Organic constructs are not built, specifically so much as they are grown. This technology is made available through advanced genetic manipulation. A creature is designed that should be able to do what is wanted in DNA and then it is simply cloned from the DNA . This can construct ships, weapons, organic robots, anything.
Things constructed this way can be built do deal with problems that other construction techniques cannot compensate for. Organic constructions can easily self heal damage, and can be made immune to certain types of energy attacks.
Organic constructs also have their own disadvantages too, in that they are vulnerable to biological attacks such as bacterial or chemical weapons.