SEO is the process of optimizing your website so that users are more likely to arrive at your site when searching for a particular word or phrase in search engines. Your site will probably come up first when searching the name of your business without much effort, but you probably want to optimize for something general like "logo design," so that users will find your site even if they haven't decided on using your business.
SEO is not an exact science because search engines do not reveal the exact algorithms for how they rank websites. It's possible that optimizing everything will still not have your website appear on the first page of search engine results. Different search engines use different algorithms, and SEO focuses most heavily on Google, which is by far the most popular search engine.
Here is a page about SEO from Google.
Use your knowledge of your business to predict what users would type into a search box when they're looking for the products or services you offer. Select one or two phrases that are of highest importance and a few other lower priority phrases. Phrases should usually be two or three words long.
For example, if your keyword phrase is "logo design," you might want your homepage title to be something like "Bob's Business Branding: Advertising and Logo Design." Not only does this improve your search engine ranking for that phrase, but it also makes users more likely to select your site when they see it on the search engine results page. However, titles should be kept to 65 characters (letters, numbers, spaces, etc.) or less; that is the largest size that will show up on search engine results pages without being cut off. The title should be different on every page.
If your keywords never appear in your site's content, search engines won't realize your site is relevant. Ideally, it's good to have 400-500 words per page, with 2 to 3 keyword phrases used 2 or 3 times each throughout the text. If you use the keyword too much, the search engine computers think your site doesn't have any actual content, but if you use it too infrequently, your site won't be associated with it effectively.
The meta tags include robots (instructions to search engines), a keyword list, and a description. These are not as important as they might seem, but you still shouldn't neglect them, especially since the description usually shows up on the search engine results page.
Many sites have a blog so that the site owner can easily add new content at regular intervals. This works especially well if you can occasionally (or even once) create entertaining or interesting content that users will want to discuss with or show to others, because the links they create will route more users to your site. Creating good "link bait" will probably require brainstorming. Here are some types of content that often encourage links:
Use paragraph tags for paragraphs, heading tags for headings, list tags for lists, etc. Your website may look the same regardless, but using correct code will help the search engines index your site. This improvement can also improve the users' experience of your site by lowering the file size (page loads faster) and making the site more accessible to people with visual disabilities.
Links to your pages improve your traffic and raise the importance of your website. The text of the link (and the text around it, to a lesser extent) is important. If the text linking to your website says "logo design," that improves your ranking for "logo design" searches. Consider using advertisements, directories, and link exchanges to have links to your sites. However, don't do link exchanges with spam sites (sites that serve little purpose except to post links to various websites to improve their Google rankings). Google's computers may think you are associated with a "bad neighborhood" and may lower your ranking or even remove you from the index entirely.