Factors:
weight of the work
material of the work: paper, wood, ceramic, metal...
material of the wall: drywall, homasote, cinderblock, brick, concrete...
appearance: flush to wall, offset from wall...
Hanging hardware
for all but the most informal and temporary installations, avoid sticky tack, adhesive tabs, or tape
for very lightweight work: tacks, T-pins, magnets
for medium weight work: sawtooth hanger, triangle-ring hanger, D-ring hanger, picture hanging wire, keyhole hanger, flush mount brackets. Affix these hangers to the work with nails, screws, epoxy...
for heavy work: french cleats (wood or metal)
On the wall:
for light work on drywall: nails
for medium weight work on drywall: drywall anchors of appropriate rating
for heavy work: screw into studs, drywall anchors of appropriate rating
for cinderblock, brick, or concrete: masonry anchors, masonry screws...
Plan ahead! Build in anchor points where you can strongly attach hangers, create a pocket where hardware can be recessed for flush hanging, use material that's thick enough to anchor a screw, etc.
French cleats are a very strong and simple method for hanging work on walls. A cleat system consists of two interlocking pieces, one attached to the wall and one attached to the work. They can be made (typically strips of wood cut with an angle on the long edge) or purchased (typically thinner versions made of bent sheet metal). If the cleat is applied to the back surface of the work, it often also requires a spacer to keep the work parallel to the wall; in some cases, you can inset the cleat so the work sits flush to the wall.
Applied cleat with a spacer to hold the work parallel to the wall.
Inset cleat set within the thickness of the piece allows it to hang flush against the wall.
Metal cleats can be thinner / lower profile than wooden ones.
Similar to a French cleat, flush mount brackets interlock with each other. Unlike French cleats, there is no side-to-side adjustability.
There are lots of off-the-shelf solutions for hanging things on the wall. Some commonly available types of hardware include sawtooth hangers, D-ring/triangle hangers, picture wire, and keyhole hangers.
Nail or screw a sawtooth hanger to the work, then hook it over a nail in the wall. Adjust the hanging position/angle slightly by moving to a different "tooth."
Nail or screw one or two ring hangers (may be triangle or D-shaped) to the work, then hook them over a nail or hook-hanger on the wall. Available in large and small sizes for light and heavy duty applications.
Nail or screw two ring hangers to either side of the work and run a piece of picture wire between them. Hook the wire over a nail or hook-hanger on the wall.
Nail or screw a keyhold hanger to the work, then hook it over a nail in the wall.
Cut a mortise or pocket for the keyhole hanger for a flush mounting system.
Using a specialty router bit, you can cut a keyhole shaped pocket directly into the work.
Hanging shelves on walls may require specialty hardware.
Choose appropriate fasteners depending on the composition of the wall and the weight of the work. Walls may be composed of drywall sheets over wooden or metal studs, plaster, concrete, cinderblock, brick, etc. Light duty hanging may require only a nail or screw. Heavier work may require locating and screwing into a stud, or using using mollies or expansion or toggle anchors.