Construction is one of the processes by which a Blockhead can reshape its world. At its simplest, construction is the placement of blocks to form floors, walls, ceilings, and other useful or decorative features.
Most blocks can be used in construction, although some are more difficult to include than others. In particular, air can't be added or removed (it automatically fills empty spaces below space), magma can't be gathered, water is not solid, and ice tends to melt into water.
Other items that can be placed are also useful in construction. Doors, ladders, and trapdoors all control or aid in easy access to different areas. For illumination, basic torches, oil and steel lanterns, and chandeliers are all effective, and windows may help during the daytime. Some gemstones also glow faintly. Crafting surfaces, chests, and beds can be both functional and decorative. Even plants and creatures can be incorporated with some forethought.
The most important purpose of construction is the creation of shelter, which improves a Blockhead's happiness by protecting it from weather and may keep it from being attacked by dropbears. A shelter can be as simple as surrounding a Blockhead with dirt blocks or as complex as a marble and gold palace floating in the air.
There are four ways to make a structure look in some particular way.
The first is the specific arrangement of blocks. Curves can only be implied, but horizontal and vertical straight lines are very easy. Leaving more space suggests roominess, which minimal space will look cramped or busy.
The second is the selection of blocks. Unless they're being removed regularly or need to resist an opposing player, there's little functional difference between most blocks, so the color and texture of the block used is more important. Exceptions include burnable blocks near a campfire, glass (which can let in light), and the ability of compost and dirt to grow plants or trees. Additionally, reinforced platforms block the formation of back walls below them.
The third is the use of paint, which allows recoloring of any of a block's surfaces.
The fourth is careful use of illumination and placeable items. Some items such as ladders are useful but can also decorate. Others such as shark jaws are purely decorative. Chandeliers are both decorative and produce powerful light. The right arrangement of light sources (and thus shadows) can alter the appearance of most blocks.
Paint deserves a particular mention. Because it can be used to recolor back walls, it should be possible for a dedicated player to make a Blockhead produce a pixel-like picture across a large expanse of back wall.