More on the shapes...

The shapes are a teaching tool. Since the Middle Ages and Guido of Arezzo, solmization syllables have been used to help singers train their voices and ears. Most people will at least recognize Do, Re, Mi, Fa, Sol, La, Ti, Do, the traditional Western solfege pattern for the major scale. These syllables are used to learn intervals and patterns in music. 

Trained musicians can apply these syllables without the aid of shapes, but around the late 1700's, shaped notes were invented so that singers would not have to understand music theory to know which syllable to assign to a given note. Each syllable is associated with a shape, and the solfege is built right in to the notation. 

The Sacred Harp uses the same idea, but with a 4-syllable system. In order to further simplify the process, a four-shape notation was adopted. A major scale would be sung Fa, Sol, La, Fa, Sol, La, Mi, Fa.

Figure from George Pullen Jackson's White Spirituals in the Southern Uplands, p. 14.

This system was introduced by William Little and William Smith in The Easy Instructor, published in 1801. As you can see, a triangle corresponds to "Fa", a circle to "Sol", rectangle to "La", and diamond to "Mi". The interval from a shape to its next occurrence in the key (Fa to Fa, Sol to Sol, etc.) is always a perfect fourth, and there is always a half-step leading to Fa (La-Fa and Mi-Fa). 

It may seem confusing at first, because of the repetition of shapes, but most singers find it very helpful once they grow accustomed to the system.