Psychometric approach
Intelligence has been historically identified by test performance. Tests may include: Give meanings of words, find analogical relations, solve new problems, speed of association, matching designs, counting backwards, etc. These test items tend to correlate with one another to a greater or lesser degree. Researchers argued that the reason that they correlated was that people had some amount of a property, intelligence, that influenced performance on almost all the items. This hidden property was first found by Spearman (1904) who invented factor analysis and ran it on test scores. He called it the general factor or simply g. He called other less general factors "s" factors. The g factor is the statistical justification for the concept of general intelligence.

Many people still believe that there is a general intelligence that people have more or less of. It is probably the modal view among professional researchers in the field. Herrnstein and Murray wrote The Bell Curve, and Jensen wrote The g Factor. Both of these large books argue that we have a single something measured by general intelligence tests that each of us has more or less of that plays a large role in classifying people. Their primary data are correlations among scores of IQ tests. They use measures of "heritability". What percent of a given trait seems to be correlated with correlated gene structure? The data supporting some view like this seem quite strong, but!  Sternberg review of the Bell Curve. I suggest that you read this discussion.

Other researchers using other principles of factor analysis came up with different analyses. One, which has had considerable influence is that of Thurstone (1938), who identified several primary mental abilities organized around content and performance areas. The ones most widely cited include: Verbal comprehension (vocabulary, verbal analogies, reading comprehension), verbal fluency (anagrams, naming words in a category, rhyming), number (speed and accuracy of simple computation), memory: (short term, strings of numbers, words or faces, note sequences); perceptual speed (proofreading, matching symbol strings, finding identical pictures); Induction (number series, word classifications, finding odd word in a group); spatial visualization (rotating figures, folding boxes)

Raymond Cattell proposed two generalized aspects of intelligence: 1) Fluid intelligence--figural analogies, series completion, and classification. 2) Crystallized intelligence: vocabulary, general knowledge, reading comprehension

Some psychometric views combine factors in a hierarchical intelligence, g, in conjunction with group and specific factors, or g combined with crystallized and fluid abilities.

One scholar, J. P. Guilford, identified three dimensions of intelligence. The different values in these dimensions can be applied across different domains. This resulted in a system with 120 different aspects to intelligence (Mayer, pp. 332-4).

Alternative structural views:
Steven J. Gould talks of two metaphors for intelligence: the ladder and the bush. The classical IQ view and cultural norms push the ladder metaphor. People even have tried to rank all animals on some unidimensional intelligence scale. Gould and many alternative views prefer the bush metaphor. The fact that an organism can perform very intelligently in one sphere says nothing about how it may perform in another. For Gould this implies that one can be intelligent in many relatively independent different ways.

Multiple Intelligences
Howard Gardner proposed different kinds of intelligences. He suggested that in order for something to be an intelligence it must meet a set of eight criteria:

Cognitive components perspective:
There are many aspects of cognition on which there are individual differences in ease of acquisition and efficiency of use. These computational components are combined and integrated to generate intelligent behavior. Robert Sternberg is the most well known proponent of cognitive components as directly involved with the nature of intelligence. He identified three categories of components: metacomponents, performance components and knowledge acquisition components. Read Sternberg on Systems approaches to Intelligence

• The Metacomponents are more like Gestalt structuring and David Marr's Computational stage of problem solving. It includes identifying the problem, selecting the appropriate representations for the problem, allocating resources, planning what to do, monitoring the ongoing process, and evaluating the results.
• The Performance Components refers to the knowledge of and the ability the actually do things, e.g. compare items, find relationships, spell words, recall words, apply effective procedures, etc. These are what are most often measured on I.Q. tests.
• The Knowledge Acquisition Components would include such things as learning vocabulary, learning how to solve logic problems, learning how to integrate polynomials, etc.