First, a definition: Wiki link: http://en.wikipedia.org/wiki/Astrobiology
The search for alien life, xenobiology, has changed a lot over the last century. In science fiction it has changed from John Carter adventuring among the various colors of “men” on a civilized Mars to Curiosity and Opportunity exploring a currently dry, barren planet surface that may have had water billions of years ago, and even more maybe, some kind of life.
The search for life on other worlds can be broken into two broad categories: searching for where humans can thrive (terraforming) and searching for what other life systems are out there (xenobiology).
At this stage it seems that carbon-based life occupies a distinct niche in the universe of life-making possibilities. It is hugely prolific in terms of both amount and variety of materials involved and the complexity of what can be created with it. There don’t seem to be any systems that are “sort of like it, but not the same”, such as silicon-based life or life with chlorine gas as the oxidizer rather than oxygen.
There may be other, way more different, styles of making life such as some kind of life living in solar plasma, but if they exist these are so different they are hard to identify and would be even harder to communicate with. Solar plasma life, for instance, would likely have a life span of milliseconds rather than years because things move around so fast and energetically in plasma.
Given all of the above big issues, where are we likely to find life we can identify?
Searching for life means searching for anomalous relations in energy flow. Example: Oxygen gas is highly reactive. It’s not going to exist for long in any environment that has large quantifies of reducing agents available, such as carbon, hydrogen or metals. The fact that Earth’s atmosphere has a lot of native oxygen in it is a sign that something is “pushing” the atmosphere and surface chemistry of Earth into an odd state, and has been doing that pushing a long time and pushing hard. That pushing is life. If we see other environments where the flow of entropy is being locally reversed in a dynamic way, as Earth’s atmosphere is, that’s a place to be looking for life. However, entropy and free energy flows are not quick and easy to measure, so this kind of research takes time.
And most life is not likely to be a prolific as Earth’s life is. The more likely version will resemble life around thermal vents deep in the oceans. It will be sparse and simple, which will make it hard to locate. Searching for life on the average planet or moon will be like prospecting for gold on Earth.
In sum, the search for xenobiology is not going to be an easy one.
For more information check out my two essays Special life-creating things about the Earth and Another Miracle of Life on Earth: Its Magnitude. Both of these are also in my Science and Insight for Science Fiction Writing book.