Most considerations of aliens consider only biological evolutionary scenarios. One considers the evolution of low life forms to the point of creating an intelligent species, and stop there. But there is no inherent barrier that stops the process of evolution once such a species is created - on the contrary, the very technology developed by it allows far greater evolutionary progress, and opens up entirely new possibilities.
It is possible for an intelligent species to manipulate its own genetic code and thus influence its own future evolution. The very beginnings of this is human breeding, or eugenics, but this was never successful over a long term. In our own era, scientific medicine allows to screen for genetic disorders and otherwise confer genetic advantages. The next step (in terms of possibility) would be the importation of genetic structures from other organisms, imported into the human genome. Further still, true genetic engineering would allow the alteration and later the creation of genetic networks in order to modify or create wholly new organs, change anatomy, and so on. Of course, one of the major goals would be the increase of intelligence and other mental faculties.
It is possible to "provolve" other species, manipulating their genetic material. This would allow, in principle, to endow other species with sapience and intelligence.
Furthermore, it is also possible to create entirely new life, based on artificial biochemistry or just on a blank-slate application of the normal biochemical and genetic structures, designing life from scratch.
In both cases, the result would be largely indistinguishable from normal evolution - the creation of a sapient intelligent species.
By far the largest area of possibility is the creation of competent artificial intelligence that will be able to compete and evolve, whether by blind evolutionary processes or more likely by artificial, recursive, intelligent redesign processes. It seems at least a plausible scenario that such self-improving intelligences would result in an exponential increase in intelligence, so that once the first such intelligence is created new and improved ones will dwarf the old ones and quickly outpace human intelligence. This is known as the "Singularity" in science-fiction and futurism circles.
It must be realized that such intelligences are still bound by physical processes. The modern personal computer is exceedingly weak (in its raw computational capacity) and wasteful (requiring a huge amount of energy and expending a huge amount of heat) in comparison with the most intelligent biological system (the human brain). In principle, however, it is possible to create far more powerful, dense, complex and efficient computational systems. Should such materials and technologies be developed, creatures of far greater intelligence are likely to be created, and creatures of human-level intelligence easily simulated and computed on such platforms. In the first stages, simply silicon and carbon compounds at normal-engineering structures can provide far more computing power than is sufficient to simulate the entire human population hundreds of times over, creating virtual worlds or robotics of 10^9 and more people-equivalents; more likely, the distribution of intelligence will by pyramidal, so that lower intelligent "individuals" will dominate, with higher-intelligent individuals and networks of individuals rarer the higher the intelligence, culminating in a few highly-trans-human intelligences.
Assuming even greater resources and technology, it is possible to convert entire asteroids, moonlets, planets, and perhaps even solar systems into "computronium", computing-matter. The information processing capacity of such megastructures is dwarfed only by the energy and industry required for their construction.
Whether through genetic engineering or artificial intelligence, eventually trans-human intelligence seems likely. Although currently a biological brain is far more effective than any artificial subtract, eventually it seems an artificial computronium would be more effective. But what would an intelligence dwarfing our own's be like? There are two possible answers to this.
Some maintain that beyond a certain level of complexity and capability, known as the Singularity, thought will be radically transformed to the point that it will be as unimaginable to us as it is impossible for an ant to imagine human thinking. In this case, wholly alien thought processes will be manifested in the computronium, patterns of computation that will lead to correct results in terms of their predictive and comprehension powers while not being reducible to logic, Turing-machines, and so on. Such a process will be "transturing". A taste of this may be Quantum Computing, which is not equivalent to a Turing machine, but quantum computing still falls short as it is equivalent to - slower - classical computation and is ultimately described by an axiomatic physical, and hence computational, theory (quantum mechanics). Another taste can be gleamed from considering Godel's incompleteness theorem; it is possible that a transhuman will utilize basic intuitions that are true but can never be proved as true based on our (limited) logical intuitions. If such trans-logical thinking is even possible is unclear. It certainly seems unthinkable from our (limited?) perspective, and hence such modes of thinking seem at least a priori impossible.
Instead, higher computational capacity would imply faster and more complex thought. A transhuman mind would be able to simultaneously contemplate a multitude of facts and factors, hold in its memory vast arrays of data, concepts, and ideas, and trivially employ complex "intuitive" thinking, each step in its thought employing complex and varied intuitions [algorithms of all sorts, possibly including quantum computing to increase calculation speed for some] rather than our own [relatively simple] logical and other intuitions. And all of this will happen rapidly, far faster than human thought.
The result of this would be thoughts which are not wholly alien. Rather, their complexity and intricacy will be such that they would be outside the reach of any human to follow. An example of this is the algorithmic solution to the Four Colors problem in mathermatics - this is solved by simple brute-force proof of all possibilities, but no human can follow all of the proofs in a human lifetime, nor hold all of the possibilities and their proofs within his head simultaneously.
To imagine a transhuman's train of thought, imagine trying to follow a book detailing it. The first thing you realize is that the text is simply huge - more than is possible to even read in a human lifetime, much less comprehend in one. Each line of thought in this massive volume is composed of a massive chain of deductions, but as you delve into them you find that each "step" is an enigma in itself, a sentence that might be true but requires difficult and inventive proof that you need to supply by yourself; for the transhuman, each step is trivially simple - for you, each is an entire intellectual project in its own right. Moving on, you realize that many of the proofs rely on considering, simultaneously, a huge number of separate ideas. You might, for example, realize that some proof requires the comparison of 100 different items, whereas us poor mortals can only consider something like 6 concepts at the same time. Further down the line, another problem creeps up - on page 239315 the text refers to theorem number 1994, and you completely forgot what it was; even going back, you realize that by the time you reached the reference you have simply completely forgotten theorem 1994, it's as if you didn't learn it. The human brain is just incapable of holding the massive amount of data and references that the text requires simultaneously. In short, it is simply impossible for a human to truly understand this tome. Even generations of humans, working in concert, will never be capable of following this text, as it is simply impossible for any one to reach the superficial level of understanding of all the others' work for the cooperation to be meaningful. The thoughts, although qualitatively still subject to human reason, have by their quantity and complexity exceeded human understanding.
Given such possibilities, it is increasingly likely that any extraterrestrial civilization we encounter will be wholly or partially virtual, with a dominant aspect of Artificial Intelligence.