Those boards are flat sawn
If you look at the ends of the boards you will see the grain is shaped like large circular segments, as it would grow in the tree...hence growth rings. As the wood dries out it tends to shrink more from the outer rings and become "cupped", a common issue when working with wide boards. That's what you have, and especially on the bottom one.
You can't prevent this, only work around it. To reduce the cup on a wide board, you simply rip it down it's length into 2 separate boards with half the amount of cup and glue them back together. Then, if you have the equipment .... you can sand them flat, plane them flat or leave them as is.
This is why plywood becomes a better choice for making cabinets. You can also use straight grained woods that have small, almost vertical lines when looking at the end grain. When these boards shrink and dry out, there is very little cupping, a better choice for making cabinet and furniture.
The best way to control any twisting or warping is to select the straightest grain boards AND to assemble them as soon as possible with whatever methods you have at your disposal... nails, screws, dados, rabbets and glue. By assembling them immediately, they will be held in place by the construction before they get a chance to warp. This also why dados are a good idea, since a board fit into a dado will not be able to cup.
If you want to make more ambitious projects, a simple entry level router with a straight edge guide or clamp would go along way to insure that your projects turn out better. Keep in mind a router is very noisy and using it in an apartment may raise the ire of nearby residents.... :frown2:
The answer to your question will only be as detailed and specific as the question is detailed and specific. Good questions also include a sketch or a photo that illustrates your issue. (:< D)