If you need Internet Explorer, you have a few options.
There's still the old, old one for Mac OS X. But you'd need a PPC based really old Mac. Like Mac OS X 10.3 or even earlier. If you have one of those around, you can install Mac OS X version of IE 5.2.2. Which is ancient and a far cry from how IE is nowadays.
But that version of IE you can probably still download from some archived software site.
Or you can use whatever you use for virtualizing or running your Windows. Boot Camp, Fusion, Parallels, Q, VPC, an old PC. You'll probably want to assume your IE viewers are using something newer than IE 7.
And then don't forget that for testing a site there are a bunch of other browsers and OS out there. Firefox, Camino, Opera, Lynx, iCab, Omniweb, Konqueror... if it's a site well made, it should work no matter what you use.
Likely that the old version 5 IE that Giaguara mentions won't help, even if it will run (because the site probably uses DirectX or .net code) and IE 6 (likely higher) is going to be needed, and that also means that you have to use some form of Windows, as Giaguara says.
Be sure to try Firefox on OS X, as that might function for your site, too.
To bolster DeltaMac's point, the most recent version of IE:mac is Internet Explorer 5.2.3. It is highly unstable and compatible with virtually no web page on the Internet today. For your personal amusement, launch IE 5.2.3. Count the seconds before it crashes.
I this day of Smart phones, tablets and other non windows internet devices there are still web developers that would exclude a huge group of users.
DeltaMac is correct if the latest Safari or Firefox will not work work with the site the old MacOS version of IE 5.2.3 sure won't
Now a days it is Really NOT Internet Explorer that is needed but Windows Internet explorer.
as Giaguara said. In that case your options are
Bootcamp and Windows : this route you need to reboot in ether windows or Mac OS you can not run both at the same time.
Parallels or VMWare Fusion VPC and Windows : allows you to run Window and Mac OS X at the same time
All of these Do require you to purchase a Licensed copy of Windows
I just find it sad as there are really only a very few reasons for developing a site today that requires Windows. And often in most cases even then there solutions out there that will do the same thing if the Windows developers only took the time to look. It is usually a Windows centric environment that has Windows Developers that do not want to take the time to learn a non windows solution.