How to determine if a compound is a strong, weak, or non-electrolyte?
It's best to start with basic definitions: electrolytes are compounds that allow a solvent to conduct an electric current. The mechanism is generally formation of ionic species, that is, solvated atoms or molecules carrying a net charge. As charge carriers, ions are the most important contributors to the electrical conductance of solutions. 

Salts represent a reliable source of ions in solution. They tend to dissociate to some extent into their component ions in solution. It is usually ok to assume that in the presence of solvent the ions in a salt can exist either as part of the solid lattice or as free ions in solution. Only the free ions contribute significantly to conductance. Therefore solubility and conductance are directly related. A salt that dissociates to a large extent is referred to as a strong electrolyte. One that does not dissociate much is called a weak electrolyte. Generally speaking the degree of dissociation depends on the lattice free energy and the free energy of solvation of the salt in the particular solvent under given conditions of temperature and pressure. Degrees of dissociation for different salts or conditions are generally tabulated as solubilities and may be described qualitatively or quantitatively (say as solubility product $K_{sp}$). Qualitative trends are usually memorized by chemists and are taught at an early level to students. While detailed explanations can usually be found for differences in solubility, they are not usually the subject of introductory courses. Memorization (and mnemonics) tend to be the rule.

Some compounds that would not generally be regarded as salts (although are sometimes prepared in salt form) contain ionizable groups that can dissociate to form oppositely charged ions. Good examples are acidic groups, such as carboxylic groups that generate an conjugate anionic (negatively charged) molecule and hydronium (in water) upon dissociation, and basic groups, such as amines, that can accept a proton from a donor compound (such as water) to form a cation (positively charged ion). Generally speaking molecular organic compounds are regarded as weak electrolytes, since the degree of dissociation of the ionizable groups may be incomplete to a large extent. This is not universally true. Some groups readily dissociate, for instance sulfates. The degree of ionization depends inasmuch on the nature of the group in question as on the structure of the compound and the solvent conditions, including in particular identity of the solvent and temperature.

Compounds that are insoluble can obviously not contribute to the conductance and are therefore not electrolytes. Some compounds readily solubilize in the solvent (for instance nonionic polar compounds in water) but do not dissociate into ionic forms. Such compounds are not electrolytes.