What exactly is IQ, and how to develop or improve it? Some people are said to have exceptionally high IQ.
What exactly is that supposed to mean?

Is it possible to develop high IQ if one doesn't have it? If yes, then how?
Literal IQ: In a literal sense, IQ is a standardised score derived from intelligence tests. Typically IQ is scaled to have a a mean of 100 and a standard deviation of 15. In that sense, it is a normative score. For children, the norm is defined relative to other children of a similar age, for adults, it is defined relative to an adult population.

IQ as g: But this still leaves the question of what an IQ score ultimately represents. At a slightly more abstract level, IQ scores typically measures something called g (perhaps g for general intelligence). In a statistical sense, g is the first component that is obtained when you perform a principal components or factor analysis of scores across a large number of cognitive ability tests. In non-statistical terms, when you get  scores on many cognitive ability tests you find that each test will typically share something with other tests, but will also contain a unique bit. g is the common bit across many tests. 

g as intelligence: Of course this still leaves the question of why there is the common variance across cognitive tests called g and whether we should care. IQ scores certainly correlate highly with educational performance, vocational outcomes, and many other social indicators (common correlations are in the .3 to .6 range).  If you are new to intelligence, a good starting point is Intelligence: Knowns and Unknowns which was put together by an APA task in order to deal with some of the controversy around IQ and intelligence. It covers conceptual definitions of intelligence, correlates and more.

Is it possible to develop IQ?
First, by IQ, I assume you mean latent IQ or latent cognitive ability. In contrast, practicing certain types of ability tests may lead to some small improvement on similar types of tests, and somewhat larger improvement on the identical test items. Cheating on a test could lead to getting a huge IQ. And all these strategies may be effective if your goal is to do well on a test perhaps for educational, vocational, or other selection purposes. However, I am assuming that you don't want to just improve on the test, you want to improve your cognitive ability in terms of the underlying ability that such tests aim to measure.

Many quick fixes to boosting intelligence are advertised under the banner such as brain training, however, there is little evidence that they are effective. And there is strong theoretical and empirical evidence that  domain specific practice does not lead to general improvements in cognitive ability. See for example these earlier questions here, here.

I tend to see boosting IQ as an unproductive goal. I think a more useful aim is to try to be good at something more specific. There is plenty of evidence that humans are able to acquire domain specific expertise with practice. IQ might help in some cases, but it is often not a major limiting factor. See the expertise literature for more information about this, from which K. Anders Ericsson is probably the most famous academic.

That said, you'll see in Intelligence: Knowns and Unknowns interesting discussions of (a) how IQ changes over childhood, (b) the relative but not complete stability of IQ in adulthood, (c) the many interventions that have been tried to improve IQ, but that generally failed to yield long term increases, (d) the mysterious Flynn effect  representing major generational increases in IQ scores, and so on. Such evidence suggests that IQ is influenced by environmental factors, but that we're not that good at understanding how to increase it. Presumably any intervention targeted at something as general as IQ would need to be similarly general, such as education or generational changes in lifestyles.