Well done search engine optimisation, often called SEO for short, is designed to allow sites to show up in search results when folks search for products rather than simply company names. The idea is that people trying to find things online will be able to search out a business website without the prior understanding of the business or its name. There are actually all manner of tricks and techniques which fall under SEO, but the good sustainable methods (also sometimes called "white hat SEO") are based around the notion of making your web page easier for search engine to read and understand, meaning they rank you higher in searches for appropriately relevant terms.
It is not uncommon to decide to outsource SEO to external
digital marketing specialists agencies and other companies, but there are far too many dodgy types available who like to take companies seeking SEO - especially local and new businesses - for a very expensive ride. How then does one actually go about finding a good, honest and legitimate search engine marketer?

It really is actually incredibly easy to seek out a quality SEO agency online. Go to a search engine like Google or Bing and type in "SEO agency." Ignoring the paid ads at the very best, whichever firm has landed at the top of the search results are what you want - they must be great at SEO or their own website would not be so high within the natural listings!
To narrow things down a bit, you could try extending your search. Say you live in Essex and want a local agency - just type in "SEO agency Essex." Or if your site is specialised and you want particular help with it you may add your industry or site type - "SEO tourism" or "SEO eCommerce" for example. This will likely be a good way to find niche SEO agencies who have specialist knowledge in your field.
Sometimes SEO agencies (or at least people claiming to be SEO agencies) try to solicit companies by sending them emails or submitting contact forms with various enticements and offers. However if you see something like this come in then it is best to ignore it - if a business which supposedly specialises in search engine optimisation is reduced to spamming inboxes to get business rather than, say, doing SEO so their very own website ranks highly and people can find them anyway, chances are they aren't very good. The exact same is true of SEO services advertised on paid advertisements at the very best of major search engines or on banner advertisements on other web sites. If their search marketing is any good, what makes them having to pay for their placements?
Another thing to bear in mind is to steer well clear of any SEO agency that promises you "top position for x keywords" or something similar. Virtually all modern major search engines used personalised searching, meaning that everybody sees websites in slightly different ranking positions depending on their personal search history. Absolute rankings are a thing of the past, any anyone claiming to deliver this kind of service is lying through their teeth. Modern SEO will use sustainable "white hat" techniques so that you can bring useful traffic to your web page, not use exploits and dodgy tricks, that may constantly get them slapped by search engine updates like Google's infamous Panda and Penguin algorithm changes.
Once you've found an SEO company that looks mostly all right you are going to need to ask them some questions prior to deciding to leap into their services. And also the common checks like existing client list and testimonials, it is an excellent idea to talk to someone directly on the phone or in person.