My website www.haveagreatdog.com has been linked to from at least 8
other significant pet websites. This has been for a period of at least
3 months. A Google search for "have a great dog" produces no results
within the first 5 pages (after that I stopped looking). My site
includes "have a great dog" in the meta tags and in the title of the
home page. It is a unique business name - shouldn't it come up first?
If you search under MY name, "David Muriello", you find many
links to my site, www.haveagreatdog.com. I can't understand why my
site won't come up in a Google search and I am extremely frustrated.
How do I ensure that www.haveagreatdog.com appears on the first page
when a user searches for "have a great dog" with Google?

Google likes words. Yet on your website the phrase "Have a great dog"
is only mentioned once in the title, once in the keywords meta tag and
once in the description meta tag.

However Google ignores meta tags as far as ranking pages is concerned.
 So the phrase is only really mentioned once on the page as far as
google is concerned.
Although you should keep the meta tags as other search engines may use them. 

You do have the phrase "Have a great dog" appear in title graphic and
in the footer graphic.  Google is unable to read text in graphics.  A
quick fix would be to add an alternative text attribute to the image
which contains the text in the graphic.

But a better option would be to replace the images with the actual
text and use style sheets to markup the text as how you want it.

Google will ignore "a", so you are left with only 3 keywords: have,
great and dog. unfortunately, 2 out of these 3 (have and great) are
very common words.

So you need more efforts to let Google rank you higher when you search
for "have a great dog".

First, replace all your footers with text. This will greatly help.

Second, since your site has a specific purpose (dog training, etc.),
register yourself at www.dmoz.org. Google and other major search
engines use this directory.

Third, try to have other sites to link to you, with your title. This
will help tremendously.