Your website should be compatible on all mobile devices regardless of the screen size or resolution. It may be that your site has been coded in such a way that it resizes all the elements correctly, but if not then you may need to either tweak the code or use a plugin to create a mobile version.
There are a number of free tools out there to allow you to check if your website is mobile compatible but some of them will only check the URL you enter, not other pages within your site, for this reason it is worth checking each page using the tool to ensure that none of them are missed.
Using tables with fixed widths causes many problems when it comes to mobile device compatibility as often they are not shrunk down with the rest of the site. If you are unsure you should check with your website developer.