There are no "best 1080p monitors" out there. Monitors are specialized and optimized for different tasks and so it all depends on so many factors, specifically on what YOU want/need as a consumer. Now when it comes to budget, I assume that if you're looking for "the best" then money should be a small obstacle.
Now there are many options to choose from, mainly these;
- TN-panels: Pros: fastest response times, most modern ones are 3D capable, cheap. Cons: bad viewing-angle, poor colorreproduction (ie. worst image quality of the lot).
- IPS-panels: basically almost the EXACT opposite of TN-panels
- VA-panels: looks to combine the advantages of both TN & IPS but does so in a "mediocre" way, leaving it somewhere inbetween the two technologies. Also, common to TN & IPS is the poor production of black images (looks almost grey). VA was introduced to address that problem.
For more reading: (http://www.tnpanel.com/tn-vs-ips-va/), (http://www.pchardwarehelp.com/guides/lcd-panel-types.php)
Now as you can see, they all have their advantages and disadv. Some say TN is the best for gaming due to its quickness, while others who may not play FPS-games but rather strategy prioritizes better image quality. So in the end, it is all up to you!