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Keyword research begins offline. Think about the words your customers might use to describe your company, as you may find the "layman" terms differ to those you and your colleagues use amongst yourselves. "SEO" itself is an excellent example of this - it is a widely searched for term amongst professionals, while "search engine optimisation" is more likely to be searched for by people who are less familiar with the subject, including those looking to hire.
Jot your ideas down then go to Google and type some in. Straight away, Google will start to suggest related terms, and you may find that many of these are also relevant to your site, so get them noted down as well. If you have Google Analytics or other keyword tracking software installed on your site, you can also make use of this data to see which keywords are currently generating the most traffic and conversions on your site.
Your next port of call is Google's Adwords Keyword tool, which will give you estimated search volumes for any keywords you enter, along with suggestions for other keywords to consider. Ensure your location is set correctly and, unless you're a global brand, take most notice of the number of local monthly searches.
[A key tip here - learn the difference between "broad" and "exact" match figures, which you can change between using the control panel on the left of the page. Simply put, broad match will give you the number of times your keyword has been entered, either by itself or as part of a longer phrase. Exact match refers only to the number of times that particular term has been entered, and so represent the number of searches for a target term much more closely.]
Now, with the raw numbers in front of you it can be all too tempting to go for the "big hitters" in your niche - keywords with tens or even hundreds of thousands of searches per month. However, consider that there are other trusted websites in your niche who may have got a head start in ranking for these keywords. Google's results pages are effectively a competition, but it's the winning that counts, not the taking part. So it's important to pick the competitions you stand a chance in, to allow you to start seeing a return on the time and financial investment you'll put into your SEO campaign.
So, consider what we in the business call "the long tail". These are the longer, lesser searched for key phrases that most accurately describe the services or products you offer, so while they may not generate as much traffic as broader terms, the visitors they do generate are more likely to be satisfied with your site as an answer to their query, and so are more likely to convert.
The best keywords find the middle ground between competitiveness, popularity and overall relevance to your site - target these and you'll take the first, solid step on the path to SEO success.