Composition is the arrangement of elements in an artwork, and the most well rendered piece in the world will still look horrible without good design.
First off, let's start with your materials. I don't mean canvas and paints, I mean the basic elements that you manipulate to create a good composition. These are the building blocks of design.
Basic Visual Elements
Line- the visual pathways that guide the eye along a work
Shape- areas defined by edges. Shapes move the eye around the composition. They can point or line up to imply lines. They can be geometric or organic (aka biomorphic). Organic shapes suggest life, and have irregular, rounded edges.
Value- relative lightness or darkness
Color- hues in various values and saturation levels. Check out the Color tutorial for details.
Texture/pattern - surface qualities that give the illusion of tactile properties
Space- the areas taken up by positive objects and negative areas
Basically, you want to decide what the focus of your artwork will be, and compose your basic design elements accordingly to create a composition that leads viewers to respond as the artist desires.
This is the concept that the elements of a design have a connection and belong together. The human mind tends to group objects into larger units. A sense of unity can be enhanced through proximity, by repeating elements throughout a composition, or by connecting forms throughout a composition with a line, edge, or direction.

Eye Magnets
Dominance of a visual element is achieved through contrast, which is the relative visual difference between one area and another. Every good 2d composition has at least one focal point, a place to which the eye is drawn. Sometimes works have multiple focal points, but if unless one is more dominant than the others this is usually detrimental to the work. It's best to make your focal point emphasized but still integrated within your work. A focal point is created by introducing variety into a unified environment.

-Places of high contrast- One thing that is different than everything else, whether in orientation, dimensionality, color, saturation, size, shape, texture, value, or whatever. 
Stability and Tension
Horizontals and verticals suggest stability.
Distortion and diagonals create tension.

Visual Weight -the higher a shape is on the picture plane, the heavier it appears to be. The darker it is, the heavier it will appear to be. Geometric shapes appear to be heavier than organic ones. Textured areas are heavier than untextured ones.
Balance and Imbalance
This is based on the distribution of visual weight within a composition.
Horizontal balance is the balance between the left and right sides of a composition.
Vertical balance is the balance between top and bottom of a composition. Our sense of gravity makes us comfortable with compositions weighted towards the bottom side of the composition.
Symmetrical balance is the simplest kind of balance, and it creates static, stable, boring compositions, usually creating a focal point along the central vertical axis.
Asymmetrical balance works by using dissimilar objects with comparable visual weights to balance a composition. A small area of dark can balance a larger light area, for instance, and a small textured area can balance a larger untextured area. Furthermore, a large element near a central vertical axis can be balanced by a smaller element closer to the edge of the composition.
Radial balance is where the compositional elements radiate out from a central point, and it can be symmetrical or asymmetrical.
In an allover pattern, there is no focal point and weight is distributed evenly throughout the composition.

Other factors to consider when building a composition:
Focus and diffusion of attention

Pattern- repetitive 2d surfaces with clear organization. The irreducible unit of a pattern is called a module, and pattern is created by repeating that module through rotation, reflection, translation and offset reflection. Repetition reduces the visual impact of a module.
Design deals with essentials. Abstraction (simplification) retains essential elements for communication. If the removal of an element does not destroy the piece, remove said element.

Rule of thirds
Placing your subject in the dead center of your work makes for boring compositions. To avoid this, artists often make use of the rule of thirds, which places focal points and other areas of interests near where lines dividing the composition into three equally large rows and columns would intersect. 

Pyramidal Compositions
Pyramidal compositions are fairly common and they're used to give an impression of stability or draw attention to a certain place. Artists construct pyramidal compositions by placing elements so that they form an imaginary triangle .

-Tools to Create an Illusion of 3 Dimensionality
-Overlapping forms

-Transparency and interpenetration of forms
-Object scale shifts- we read objects as the same thing at different distances
-Position- things farther down on the picture plane appear closer to the viewer
-Receding forms- for instance, a trapezoid can appear to be a rectangle that is receding in space
-Fading. high contrast forms appear closer: atmospheric perspective aka aerial perspective refers
to the idea of things fading as they recede in space. Think of a distant mountain range; landscapes
also usually take a tinge of the sky color as they recede.

-Foreshortening- a form of close up, amplified perspective
-Variations in line quality- thick lines read as near, while thin lines read as far away.
-Perspective- this includes elements of all of the above.