Ad servers enable affiliates to plug in a short bit of code to places on their site(s), and then easily change out banners, as well as rotating a variety of banners.
Using an ad server also simplifies the management of affiliate campaigns, because you can see and compare all of your activity in one place.
I used a service called OpenX for years.
It's free, but it also has a bit of a learning curve, and probably has more options than the average affiliate needs.
Google also rolled out Google Ad Manager back in 2008, but it seems to have been phased out at some point.
Since you're using WordPress for your affiliate site, I would encourage you to go with a more seamless method to manage your ads.
There are countless WordPress plugins for managing ads. Just go to the Plugins section on your WordPress dashboard and click “Add New”. This will enable you to search for new plugins.
Then search for ad manager or ad rotator, and try out one of the higher rated plugins.
Note that some are free and others cost something.
I haven't used a plugin for ad management, yet, so there isn't one in particular that I would recommend. Just try a few out and stick with the one you like the most.