Due to my Cognitive Science background I have a tendency to regard intelligence from an information processing perspective. Of course intelligence is a very general term, but it makes sense to have some sort of criterion before one begins to dispute related issues. Thus Cognitive Scientists judge a systems intelligence through its ability to efficiently process information in response to a given task or problem.
The issue here is whether intelligence is inherited genetically or gained from the environment through experience. I will focus on human intelligence, although it equally applies to any animal with a brain. Furthermore I will ignore genetically inherited diseases that effect brain performance (e.g. Down syndrome), and concentrate on normally functioning brains.
In short I will show that both genetics and experience have a role to play in our intelligence.
The human brain consists of several 100 billion interconnected neurons that receive information from the environment through our sense organs, process information, and then send signals to our motor neurons so we can respond with intelligent and appropriate behavior. These neurons are organized into neural networks and their patterns of electrochemical activation are causally responsible for cognition, consciousness and the intelligent behavior that every one of us displays. There are around 100 different neural networks in the human brain, each performing a particular function (although these networks are by no means independent).
Our inherited genetics in part encode the size and physical structure of our neural networks during development before and after birth. Other factors such as provision of correct nutrients during gestation, and the correct diet during brain development after birth, also affect the structure; but a large part can be put down to genetics. Bigger neural networks increase information processing capacity by increasing processing power and memory (which is stored in neurotransmitter concentrations in the neural cell bodies). Hence people who inherent big brains tend to be more intelligent. However before a human experiences the world and provides real world informational input to their neural networks, much of the organizational structure of their networks is chaotic. Hence until babies' brains (and bodies) develop, they are unable to understand very much or perform anything intelligent.
As an infant's neural networks are fed input through their sense organs, their networks become more organized and more efficient at processing information. Repeated exposure to certain experiences allows certain skills to be developed. From motor skills such as walking, to linguistic skills, to logic skills; all of which contribute to intelligent behavior. This is due to the developing neurological organization of the networks, which allow them to process and store information in a more efficient way.
D.Hebb (The Organization of Behavior: A Neuropsychological Theory, (New York: Wiley 1949), p.62):
"When an axon of cell A is near enough to excite a cell B and repeatedly or persistently takes part in firing at it, some growth purpose or metabolic change takes place in one or both cells such that A's efficiency, as one of the cells firing to B is increased."
Hence by being exposed to certain tasks/problems in our experiences we become better at processing related information, and can be regarded as more intelligent in that domain. For example expert chess players reach such an intelligent level of expertise due in large part to the fact they have played a lot of chess, and are thus able to recognize certain patterns from previous experience when they play a game and apply the correct moves.
Artificial neural network simulations have been computer programmed to demonstrate this ability of neural networks to increase IP efficiency following repeated exposure of input. For example Sejnowski and Rosenberg's NETtalk (1987), which contained a huge set of parallel artificial neural networks, learnt to read and correctly articulate English text phonetically through trained examples. Following network training, results of the phonetic output started from random noises, through to babbling like a baby, to a fairly high competence in word articulation of about the level of a four year old. Of course every artificial neural network simulation programmed so far is nowhere near the complexity of the human brain's neural networks; but they do provide evidence that the theory works. With the exponential growth rate of software complexity and computer IP power we are bound to see more realistic simulations in the future.
In conclusion then, evidence from Cognitive Science, Artificial Intelligence, and Developmental Neuroscience provide strong indication that our intelligence is in part inherited from our genetics, which encode a certain structure and size of our neural networks. But it isn't until we experience the world and the networks are given informational input, that we really develop any true intelligence. Hence a person disposed to the correct upbringing, education and general environment will be far more likely to develop intelligence, than someone less so disposed. Clearly then both genetics and environment are important to human intelligence, but environmental upbringing is the key.
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