When designing and coding a website, what browsers should I use to see if it display correctly?

I have worked with about 3-4 browsers when coding a website, and in all of them it display in a different way.
What should I use for this?

You should make sure your pages display correctly on all browsers. Otherwise you may lose or discourage over 50% of your visitors.

Make sure you use XHTML1.0 Compliant code on your pages. If you are using a coding software such as Dreamweaver use the CSS module rather than table-based design to enforce compliance across all of your design attributes.

Some sites place a note saying, "Best viewed with Firefox".

That's because FireFox outputs all XHTML1.0 Compliant tags, where as IE misses some attributes that should be outputted, causing minor display errors. I think FireFox is best too, but a lot of people still use IE so it's best to check your output on both.

most software developers know that firefox has surpassed what IE is capable of... Firefox has been sited as one of the samples that could be achieve when we speak of free and opensource community.

apparently, most end-user doesn't know anything about this. IE7 is actually a far beyond what Firefox can do.

As such, most developers having the desire to spread out the news uses such tags to let the non-techy end-users know that they can be served well with firefox.